What evidence do we possess today indicating that Ignatius of Antioch was ordained by the Apostles?
It appears we have no hard evidence of this, in spite of the legends that have grown up around Ignatius of Antioch. Thomas J. Heffernan says in Sacred Biography, page 58, that we know nothing about the historical figure of Ignatius of Antioch other than his journey to martyrdom from Antioch to Rome and that he wrote letters to Christian communities along the way. Allen Brent says in Ignatius of Antioch page 14, that Ignatius tells us nothing about his life prior to his departure from Antioch in chains. In other words, we do not know when Ignatius was born, in spite of the persistent tradition that he was born in 35 CE, and we do not know when he was ordained or by whom.
Ignatius died in 107 CE. At a time when commoners rarely lived beyond about 50 years, the tradition that he was born in 35 CE means that he was executed at the age of 72, after withstanding the rigours of captivity that would have killed a much younger man. There being no evidence that he was really born in 35 CE, f we make a different guess at his date of birth, say 57 CE, and assume that he was ordained at the age of 20, then his ordination as a deacon would have taken place around 77 CE - after Peter and the other apostles were probably all dead.
Wikipedia states that Theodoret of Cyrrhus reported that St. Peter appointed Ignatius to the episcopal see of Antioch. However, Theodoret lived during the fifth century, centuries after the time of Ignatius, and could not be regarded as a good historical source. Perhaps Ignatius was ordained by an apostle, but we have no way of knowing.
Whenever you have questions like this, the first place to turn is wikipedia. Of course, you can never really "trust" wikipedia, but you can still get a good overview of the situation, and---much more importantly---you can use its references to track down the information you really want. So, if you visit the wikipedia page on Ignatius, you will find two primary sources. I have tracked them down for you. The first is Theodoret, who wrote in Dialog 1---the immutable:
You have no doubt heard of the illustrious Ignatius, who received episcopal grace by the hand of the great Peter, and after ruling the church of Antioch, wore the crown of martyrdom.
Now concerning those bishops which have been ordained in our lifetime, we let you know that they are these:---James the bishop of Jerusalem, the brother of our Lord; upon whose death the second was Simeon the son of Cleopas; after whom the third was Judas the son of James. Of Cæsarea of Palestine, the first was Zacchæus, who was once a publican; after whom was Cornelius, and the third Theophilus. Of Antioch, Euodius, ordained by me Peter; and Ignatius by Paul.
So, there appears to be conflicting evidences, and both very late (fourth and fifth centuries CE).
According to wikipedia, Eusebius also mentions Ignatius. This has, oddly enough, proved more difficult for me to track down. But you can explore the references---and pretty much everything by Eusebius has a free online English translation available.
But then, according to the post-Nicene fathers, we should consult the so-called Dictionary of Christian Biography and Literature to the End of the Sixth Century for more information. There---in particular, in the Euodius/Evodius entry, we find the following very rich information:
Evodius (1), according to early tradition, first bp. of Antioch (Eus. Chron. ann. Abr. 2058; H. E. iii. 22). His episcopate has indirectly the older testimony of Origen, who speaks of Ignatius as the second bishop after Peter (in Luc. Hom. 6, vol. iii. p. 938; see also Eus. Quaest. ad Steph. ap Mai, Scr. Vet. i. p. 2). This tradition has all the appearance of being historical. Ignatius early acquired such celebrity that it is not likely the name of an undistinguished person would have been placed before his, if the facts did not require this arrangement. The language used about episcopacy in the Ignatian epistles agrees with the conclusion that Ignatius was not the first at Antioch to hold the office. As time went on, the fitness of things seemed to demand that Ignatius should not be separated from the Apostles. Athanasius (Ep. de Synodis, i. 607) speaks of Ignatius as coming after the Apostles without mention of any one intervening; Chrysostom makes him contemporary with the Apostles (Hom. in Ignat. vol. ii. p. 593); the Apostolic Constitutions (vii. 46) have recourse to the expedient adopted in the parallel case of Clement of Rome, the hypothesis of a double ordination, Evodius being said to have been ordained by Peter, Ignatius by Paul. Theodoret (Dial. I. Immutab. iv. 82, Migne) and others represent Ignatius as ordained by Peter. The authorities are given at length by Zahn (Patres Apostol. ii. 327). There is reason to believe that the earliest tradition did not include an ordination even of Evodius by Peter; for the chronicle of Eusebius places the departure of Peter from Antioch three years, or, according to St. Jerome's version, two years before the ordination of Evodius. The chronology of the early bishops of Antioch has been investigated by Harnack (Die Zeit des Ignatius). He infers that the earliest list must have contained only names of bishops of Antioch without any note of lengths of episcopates, but still that Eusebius must have had the work of some preceding chronologer to guide him. We may well believe, as Harnack suggests, that Eusebius got his chronology of early bishops of Antioch from Africanus, to whom he acknowledges his obligation, and whose chronicle has generally been believed to be the basis of that of Eusebius. If the belief had been entertained at the beginning of the 3rd cent. that Evodius had been ordained by Peter, it is incredible that Africanus would have assigned a date which absolutely excludes an ordination by Peter. The date assigned by the chronicle of Eusebius to the accession of Evodius appears to have no historic value, and thus, while we accept the episcopate of Evodius as an historic fact, we have no data for fixing his accession, but may safely place it considerably later than a.d. 42.
This is, obviously, full of references. But the abbreviations are hard to decode, and so I leave it to you to investigate further.