Sola Scriptura is most often defined as the idea that the Bible is the sole infallible source of authority for Christian faith and practice.
Do Sola Scriptura adherents claim a Biblical Basis for their belief and if so, what is it?
"It is written...", declared Jesus when standing against the temptations of Satan in the wilderness. It is strong evidence when the Lord of Glory himself, in resisting the onslaught of the wicked one, resorts not to tradition nor natural law nor human reason nor his inherent ability to cast out Satan but to what is written.
In Mark 7 Jesus chastises the Scribes and Pharisees for elevating tradition above the written commands of God:
And he said to them, “Well did Isaiah prophesy of you hypocrites, as it is written, “‘This people honors me with their lips, but their heart is far from me; in vain do they worship me, teaching as doctrines the commandments of men.’ You leave the commandment of God and hold to the tradition of men.” And he said to them, “You have a fine way of rejecting the commandment of God in order to establish your tradition! - Mark 7:6-9
Notice that he does not condemn tradition in and of itself but tradition as it becomes equal or greater (in the eyes of men) than what is written. This is not the only instance where such chastisement occurs.
Often, when questioned, Jesus would refer the questioner to the Scriptures:
And behold, a lawyer stood up to put him to the test, saying, “Teacher, what shall I do to inherit eternal life?” He said to him, “What is written in the Law? How do you read it?” - Luke 10:25-26
Other times he would expound upon the Scriptures or rightly interpret them to his audience and, in doing so, directly refuted tradition and the prevailing thought of the day:
But Jesus answered them, “You are wrong, because you know neither the Scriptures nor the power of God - Matthew 22:29
And often greatly angered them in doing so:
And he said, “Truly, I say to you, no prophet is acceptable in his hometown. But in truth, I tell you, there were many widows in Israel in the days of Elijah, when the heavens were shut up three years and six months, and a great famine came over all the land, and Elijah was sent to none of them but only to Zarephath, in the land of Sidon, to a woman who was a widow. And there were many lepers in Israel in the time of the prophet Elisha, and none of them was cleansed, but only Naaman the Syrian.” When they heard these things, all in the synagogue were filled with wrath. And they rose up and drove him out of the town and brought him to the brow of the hill on which their town was built, so that they could throw him down the cliff. - Luke 4:24-29
Suffice it to say that Jesus' "go to" was the written word of God over and above any other resource and this is a salient point, indeed.
The Apostle Paul is another great example. He was called and specifically gifted by the Lord Jesus Christ to bear His name before the Gentiles. He was given great authority and power to preach the Gospel, establish churches, teach doctrine, and rebuke false teachers. Many of his letters are included within the canon of Scripture and his writings were acknowledged by his contemporaries as Scripture long before the idea of "canon" was conceived:
And count the patience of our Lord as salvation, just as our beloved brother Paul also wrote to you according to the wisdom given him, as he does in all his letters when he speaks in them of these matters. There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures. - 2 Peter 3:15-16
And yet this man, with Apostolic authority to actually write Scripture, preached to those in Berea who, when they heard his message, diligently searched the Scriptures to determine whether what Paul was saying was true:
And the brethren immediately sent away Paul and Silas by night unto Berea: who coming thither went into the synagogue of the Jews. These were more noble than those in Thessalonica, in that they received the word with all readiness of mind, and searched the scriptures daily, whether those things were so. - Acts 17:10-11
This preaching was done in the synagogue of the Jews, a place steeped in tradition elevated above and added on to what is written, and the author of Acts pointedly remarks on the nobility of their actions in verifying Paul's preaching in Scripture rather than tradition. Because they discovered Paul's preaching to be rooted in Scripture many believed.
There are many things which could have been written. What has been written has been written for specific purposes in the eternal counsel of God:
And many other signs truly did Jesus in the presence of his disciples, which are not written in this book: But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name. - John 20:30-31
Tradition, doctrine, and entire theological schools of thought may be derived from Scripture and informed by Scripture. None of these should ever be used to inform Scripture. There is so much more that can be pulled from the pages of Scripture but, fundamentally, this is the biblical basis of Sola Scriptura:
It is written...
There are several more scriptures that point to the infallible nature of God's word.
Jesus said "The scripture cannot be broken" John 10:35
Peter said that the written word of God is "more sure" compared to the direct voice of God from Heaven.
KJV 2 Peter 1:18-19 And this voice which came from heaven we heard, when we were with him in the holy mount. We have also a more sure word of prophecy; whereunto ye do well that ye take heed, as unto a light that shineth in a dark place, until the day dawn, and the day star arise in your hearts:
KJV Isaiah 8:20 To the law and to the testimony: if they speak not according to this word, it is because there is no light in them.
Anyone speaking against the written word of God has no light in them.
Psalms say that God has magnified His word above all His name:
KJV Psalms 138:2 I will worship toward thy holy temple, and praise thy name for thy lovingkindness and for thy truth: for thou hast magnified thy word above all thy name.
And as it has been rightly pointed out, God cannot lie(Heb 6:18) neither can He change (Mal 3:6), this assures us of the truthfulness of His word.
A curse has been pronounced in anyone trying to change the word of God:
KJV Revelation 22:18-19 For I testify unto every man that heareth the words of the prophecy of this book, If any man shall add unto these things, God shall add unto him the plagues that are written in this book: And if any man shall take away from the words of the book of this prophecy, God shall take away his part out of the book of life, and out of the holy city, and from the things which are written in this book.
This shows that God is very serious when dealing with His word and we can safely trust the word
I would agree with the OP definition of Sola Scriptura (SS), but here is a bit more. From there, I will show that both the Old and New Testaments rely on and prove this.
Sola scriptura means that Scripture alone is authoritative for the faith and practice of the Christian. The Bible is complete, authoritative, and true. What is sola scriptura? | GotQuestions.org
Sola scriptura, meaning by scripture alone, is a Christian theological doctrine held by some Protestant Christian denominations, in particular the Lutheran and Reformed traditions of Protestantism,1 that posits the Bible as the sole infallible source of authority for Christian faith and practice. Sola scriptura - Wikipedia
Sola Scriptura declares that only Scripture is our inerrant, sufficient, and final authority for the church, because it is God breathed and divinely inspired (2 Timothy 3:16). What Is Sola Scriptura? | Zondervan Academic
By sola Scriptura Protestants mean that Scripture alone is the primary and absolute source for all doctrine and practice (faith and morals). What Is Sola Scriptura? | Christian Research Institute (equip.org)
In short, SS means that the bible alone is the source for all things salvific (doctrine and practice).
Although it may seem strange, I will start by backing into the SS belief from early tradition. The point is tradition may be useful, but it is not authoritative, binding the believer. In other words, was SS a Reformation idea or has it existed from the time of Peter, Paul, James, John, and the others?
(73) But Hermas wrote the Shepherd (74) very recently, [7c] in our times, in the city of Rome, (75) while bishop Pius, his brother, was occupying the [episcopal] chair (76) of the church of the city of Rome. [7d] (77) And therefore it ought indeed to be read; but (78) it cannot be read publicly to the people in church either among (79) the Prophets, whose number is complete,  or among (80) the Apostles, for it is after [their] time. Muratorian Fragment
The Muratorian Fragment defines the conditions for the writings of the Old Testament, a valid prophetic line, and for the New Testament, during the time of the apostles. Other writings, other opinions after the time of apostles (James to John sons of Zebedee, first and last apostles to die) may be of interest and may be pious, but they are not authoritative for the church.
The Muratorian Fragment was written circa 175 CE. A mere 10 years or so later Polycrates will also search scripture alone for his beliefs and practices.
- I, therefore, brethren, who have lived sixty-five years in the Lord, and have met with the brethren throughout the world, and have gone through every Holy Scripture, am not affrighted by terrifying words. For those greater than I have said ‘We ought to obey God rather than man.’” -source-
Was this tradition made up? Absolutely no, we find it in Scripture, the source to what the Fragment is referring.
Much every way: chiefly, because that unto them [the Jews] were committed the oracles of God. Romans 3:2
The word “committed” means to entrust, to be persuaded of. G4100 - pisteuō - Strong's Greek Lexicon (kjv) (blueletterbible.org)
“Oracles of God” means a diminutive of logos, "a word, narrative, statement," denotes "a Divine response or utterance, an oracle;" Vines https://www.blueletterbible.org/lexicon/g3051/kjv/tr/0-1/
As Hebrews 1:1 puts it, “Long ago, at many times and in many ways, God spoke to our fathers by the prophets,”. God spoke, not man made-up.
And Peter 2 Peter 1:21 For no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit. From God. And Paul Acts 13:32, And we bring you the good news that what God promised to the fathers,. God promised. God.
And Jesus in many places confirms this; it is written.
Lastly the noble Bereans would search for what is written in the Scriptures to see if it was true (Acts 17:11). True, from God or not.
Paul again from Romans 15:4 For whatsoever things were written aforetime were written for our learning, that we through patience and comfort of the scriptures might have hope.
Note that in all of these places, no quarter is given to the traditions or thoughts or ruminations of men.
So, the Old Testament was written during the valid prophetic line. Thus, it is the word of God.
What of the New Testament?
The same theme of the authoritative and sufficient “it is written” of prophets and apostles continues.
That ye may be mindful of the words which were spoken before by the holy prophets, and of the commandment of us the apostles of the Lord and Saviour:. 2 Peter 3:2 The same level of authority between prophets and apostles is shown.
And are built upon the foundation of the [NT] apostles and [OT] prophets, Jesus Christ himself being the chief corner stone; Eph 2:20
It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus, That thou mightest know the certainty of those things, wherein thou hast been instructed. Luke 1:3-4 Writing it down made it certain, rather than guessing.
But if ye believe not his writings, how shall ye believe my [Jesus] words? John 5:47
But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name. John 20:31
And the gospel must first be published among all nations. Mark 13:10
Beloved, when I gave all diligence to write unto you of the common salvation, it was needful for me to write unto you, and exhort you that ye should earnestly contend for the faith which was once delivered unto the saints. Jude 1:3
These things have I written unto you that believe on the name of the Son of God; that ye may know that ye have eternal life, and that ye may believe on the name of the Son of God. 1 John 5:13
Contend for the faith. This would be meaningless unless the faith had been delivered via writing it down. To be clear, the apostles first spoke, but later wrote it down.
Along those lines, what those who do not believe in SS must believe (and prove) is that the apostles left something out, that God forgot to tell them something necessary for our salvation. But as it is written Jesus saying, But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you. John 14:26
So, to answer the OP, yes, Sola Scriptura adherents do have a biblical basis for our belief.
Not only is there biblical basis for Sola Scriptura, there is no other basis than the Bible! Many biblical texts have already been quoted in other answers, but one that bears repetition was written to Christians living prior to AD 65 or, at the latest, around AD 80. They were told to
"contend earnestly for the faith that was once for all delivered to the saints" (Jude vs. 3).
Then they were warned about men who had already crept into the congregation with heretical teachings and immoral practices. Jude is reminding Christians that 'the faith' had already been given to the Church. If they stuck to that, they would not be tricked by those pseudo-Christians. Given that the only book of scripture that had not yet been written was the Revelation prophecies about future events leading up to Christ's return, which did not add any new teaching to the Christian faith, all that the apostles had taught about the faith to be believed was already committed to writing.
Here is a list of other Bible texts upon which Sola Scriptura is based.
O.T. Psalm 119:129-133, 160 & 138:2; Isaiah 8:20; Daniel 9:24
N.T. Matthew 22:29; Mark 7:6-9; John 5:47: 10:35: 14:26: 20:31; Acts 17:11; Romans 15:4; Ephesians 2:20; 2 Timothy 3:16; Hebrews 1:1-3; 2 Peter 1:18-19 & 3:2; 1 John 5:13; Revelation 22:18-19.
"What is written" and "the scriptures say..." is of supreme importance and necessary for salvation. When the Reformed faith talks about Sola Scriptura, it is talking about how the Bible (holy scripture) stands alone among books (religious and/or secular) as God's authoritative word given by him to men. Three other things flow from this:
(1) The Bible is a perfect revelation of God's mind for man's salvation. He didn't cause everything he knows to be written down; nor did he have written down everything in world history. He caused to be written down all that was necessary for us if we are to know eternal life in God and Christ (John 17:3).
(2) The Bible is sufficient revelation. It is enough. Religions other than the Christian one that believes in Sola Scriptura appear to need additional (new) writings or prophets to explain "what the Bible really means".
(3) The Bible is final revelation. Messiah would "seal both vision and prophet" (Dan.9:24). That is why the Book of the Revelation, given by Christ to the last remaining apostle circa AD 95 concludes the Bible with visions of events leading up to his still-to-come return.
Many of these points gleaned from an article in "The Monthly Record" March 2014, "Solas - article 2 of 6'.
Much of this depends on the definition of Sola Scriptura. The most accurate an agreed upon definition is this: Scripture alone is infallible and authoritative for the faith and practice of the Christian.
Sola Scriptura adherents will find much of the support for their position from 2nd Timothy 3:16-17:
16 All Scripture is breathed out by God and profitable for teaching, for reproof, for correction, and for training in righteousness, 17 that the man of God may be complete, equipped for every good work.
The analyzation of this passage is as follows: Since God cannot lie according to Hebrews 6:18, and knows all according to 1 John 3:20 and any scripture is an extension of God's character (since its breathed out by God, thus God is speaking through the text), the scripture must be infallible (without error).
Since God is also the key source of authority according to Romans 13:1, the scriptures must also be authoritative.
Then there is the question of scriptures being the sole infallible and authoritative rule of faith.
SS defenders find that from verse 17, that man is "complete, [fully] equipped for every good work." I include the bracketed fully, as some translations do include that.
So that's the basic argument from 2nd Timothy. Other scriptural supports include the Bereans being commended for testing doctrine against the scriptures at Acts 17:11, and in 1 Corinthians 4:6, where Paul says to not go beyond what is written. Other smaller arguments may come from Jesus condemning the people who went to tradition rather than the law of God.
I'd also like to add this - Protestants would claim that since no explicit command of another source of authority is found in the text, we should only go to the scriptures as it's all we can trust.
As a final note, I'd like to point out the necessity of there being scriptural backing to Sola Scriptura. If there is no scriptural backing, then Sola Scriptura is self refuting. Every doctrine should be found in the scriptures according to Sola Scriptura, which includes the doctrine of Sola Scriptura.