There is Biblical Support
While, as I will later explain, there is no and can be no basis in the Bible for such a critical part of Christian life, there is nonetheless an abundance of passages that can be read to present tensions between the belief that the Bible contains the Inspired Word of God and that such practice and teaching is not consistent with the will of God.
In 2 Corinthians 2:10 Paul talks about performing the Sacrament of Reconciliation "in persona Christi" (in those words, in the Latin), which provides clear support for the idea that people can do this. And then the typical support for the authority of the priesthood could be read as support that it is the priests who do this.
Furthermore, and luckily for you, Pope Benedict XVI in his Apostolic Exhortation Sacramentum Caritatis, focusing broadly on the Eucharistic Sacrament, devoted an entire subsection to the role of the Sacrament of Ordination as relates to the Eucharist, and a subsection thereof titled "In Persona Christi Capitis" which is, I think, exactly what you are looking for. It explains the Church's position on this, with copious Biblical citations, among others. I have reproduced it below, from the web documents of the Holy See.
The intrinsic relationship between the Eucharist and the sacrament of Holy Orders clearly emerges from Jesus' own words in the Upper Room: "Do this in memory of me" (Lk 22:19). On the night before he died, Jesus instituted the Eucharist and at the same time established the priesthood of the New Covenant. He is priest, victim and altar: the mediator between God the Father and his people (cf. Heb 5:5-10), the victim of atonement (cf. 1 Jn 2:2, 4:10) who offers himself on the altar of the Cross. No one can say "this is my body" and "this is the cup of my blood" except in the name and in the person of Christ, the one high priest of the new and eternal Covenant (cf. Heb 8-9). Earlier meetings of the Synod of Bishops had considered the question of the ordained priesthood, both with regard to the nature of the ministry (69) and the formation of candidates.(70) Here, in the light of the discussion that took place during the last Synod, I consider it important to recall several important points about the relationship between the sacrament of the Eucharist and Holy Orders. First of all, we need to stress once again that the connection between Holy Orders and the Eucharist is seen most clearly at Mass, when the Bishop or priest presides in the person of Christ the Head.
The Church teaches that priestly ordination is the indispensable condition for the valid celebration of the Eucharist.(71) Indeed, "in the ecclesial service of the ordained minister, it is Christ himself who is present to his Church as Head of his Body, Shepherd of his flock, High Priest of the redemptive sacrifice." (72) Certainly the ordained minister also acts "in the name of the whole Church, when presenting to God the prayer of the Church, and above all when offering the eucharistic sacrifice." (73) As a result, priests should be conscious of the fact that in their ministry they must never put themselves or their personal opinions in first place, but Jesus Christ. Any attempt to make themselves the centre of the liturgical action contradicts their very identity as priests. The priest is above all a servant of others, and he must continually work at being a sign pointing to Christ, a docile instrument in the Lord's hands. This is seen particularly in his humility in leading the liturgical assembly, in obedience to the rite, uniting himself to it in mind and heart, and avoiding anything that might give the impression of an inordinate emphasis on his own personality. I encourage the clergy always to see their eucharistic ministry as a humble service offered to Christ and his Church. The priesthood, as Saint Augustine said, is amoris officium, (74) it is the office of the good shepherd, who offers his life for his sheep (cf. Jn 10:14-15).
There is no Biblical Basis
The Church does not base Her theology on the Bible. That would be nonsensical, seeing as the Church predates the Bible by several centuries and, indeed, several books of the Bible codify the teachings of persons already acting within the forming hierarchy of the early Church. We read about people playing at priesthood to ill effect in Acts 19:13-16, and the Didache was published earlier than the Pauline epistles, and already assumes its readers will possess orthodox theology from other sources. The Bible came into existence long after the Church had finished Her formation/reformation following the Ascension of Christ.
I mention this to point out that it is rather silly to ask for a Biblical Basis for Catholic doctrine, since most Catholic doctrine predates the Bible and for any such doctrinal teachings one would need to assume retrocausality to arrive at such a basis. In particular, essential ministries such as the Eucharist have been practiced with unchanging essence since the time of the Apostles. The form of the Mass has evolved quite dramatically (early services involved a play, took place in secret in forbidden crypts, and were otherwise very different. You can look at Orthodox and Catholic but not Roman services to see many other valid ways of doing things that also starkly contrast with one another as well as those of churches using the Latin Rite), but the "important parts", if you will, have remained the same. The consecration of the host and receipt of the Eucharist is the whole focal point of a Mass, and so it should be no surprise that said consecration predates the Bible by four centuries or so.
In any case:
1) This Catholic doctrine predates the Bible and so cannot be based on it
2) Your statement of this doctrine is wrong; that is not quite what the Church teaches on the matter. You can read the Church's actual position from the Cathechism (relevant subsection linked). I recommend you begin reading at passage 1548.
3) The teachings of the Church on the matter are supported, in a general sense, by the same Biblical material that generally supports the institution of the priesthood itself, as well as those that generally support the particular essential elements of the Eucharistic Sacrament.
