Is there any evidence to support the claim that the Apostle St. Peter founded the Church in Antioch and, if so, what are the implications? According to tradition, the Apostle Peter founded the Church in Antioch in A.D. 34 and is regarded as the first Bishop of Antioch.  Here are a few extracts from  sources I found regarding this claim:


  Peter preached the Gospel in Judea, founded the Church of Antioch and finally came to Rome... St. Peter traditionally is regarded as the leader of the Twelve Disciples of Jesus... In the course of his missionary journeys, Peter founded the Church in Antioch, where the followers of Jesus were first called Christians. St. Peter is regarded by the Church as the first Bishop of Antioch, and the present-day Patriarch of Antioch is his successor in that Apostolic See. - Saints Peter and Paul
  
  Church tradition maintains that the See of Antioch was founded by Saint Peter the Apostle in A.D. 34... It was the resolution of this conflict [male circumcision] at the Council of Jerusalem under Saint James the Apostle that determined the direction of the Antiochian mission to the Gentiles, and the dynamic nature of that Christian community in its missionary outreach. - The Patriarchate of Antioch: Founded by Saints Peter and Paul
  
  Patriarch of Antioch is a traditional title held by the Bishop of Antioch (founded by Saint Peter). As the traditional "overseer" (ἐπίσκοπος, episkopos, from which the word bishop is derived) of the first gentile Christian community, the position has been of prime importance in the church from its earliest period. This diocese is one of the few for which the names of its bishops from the apostolic beginnings have been preserved...  According to church tradition, this ancient Patriarchate was founded by the Apostle Saint Peter. - Patriarch of Antioch (Wikipedia) 


Apparently Ignatius was bishop of Antioch (martyred under Emperor Trajan circa A.D. 110)

Theophilus was bishop of Antioch (A.D. 169-183)

Serapion was bishop of Antioch (A.D. 190-203)

Is there any substantive evidence to support the claim by the Antiochian Orthodox Church that the apostle Peter founded their church?  And, if so, what are the implications with regard to the Roman Catholic Church claiming Peter founded their church in Rome?
Interesting question akin to did Peter found the church at Rome.  Historical evidence of either is scant, but here is Eusebius and Schaff's commentary.  For Eusebius, Evodius was first bishop, followed by Ignatius.


  At this time Ignatius was known as the second bishop of Antioch, Evodius having been the first.
  -source-


And commentary:


  738    We cannot doubt that the earliest tradition made Evodius first bishop of Antioch, for otherwise we could not explain the insertion of his name before the great name of Ignatius. The tendency would be, of course, to connect Ignatius directly with the apostles, and to make him the first bishop. This tendency is seen in Athanasius and Chrysostom, who do not mention Evodius at all; also in the Apost. Const. VII. 46, where, however, it is said that Evodius was ordained by Peter, and Ignatius by Paul (as in the parallel case of Clement of Rome). The fact that the name of Evodius appears here shows that the tradition that he was the first bishop seemed to the author too old and too strong to be set aside. Origen (in Luc. Hom. VI.) is an indirect witness to the episcopacy of Evodius, since he makes Ignatius the second, and not the first, bishop of Antioch. As to the respective dates of the early bishops of Antioch, we know nothing certain. On their chronology, see Harnack, Die Zeit des Ignatius, and cf. Salmon’s article Evodius, in Smith and Wace’s Dict. of Christ. Biog.
  -ibid-


So, according to the Apostolic Constitutions Book VII P 46, the claim is that Evodius was first bishop, but who was appointed as such by Peter.


  Of Antioch, Euodius [Evodius], ordained by me Peter; and Ignatius by Paul. 
  -source-


The sense is not necessarily that Peter lived for years in Antioch, but rather visited and established his successor Evodius, the first bishop.

Implications of this tradition of Peter founding the church at Antioch?  It depends on one's interpretation of the well discussed "upon this rock" being either Peter or his confession.  If one believes the rock refers to Peter, then we have competing claims as to which church that might be; that is, either the one sourcing to Antioch or Rome or even Jerusalem.  If one believes the rock refers to one's confession of faith in Christ, then it basically does not matter the location of where Peter roamed, but rather just the facts that he and the other apostles did carry the gospel, the good news, message outward from Jerusalem to the ends of the earth.