Is there any possible evidence to support the claim of the Antiochian Orthodox Church that their Church was founded by the Apostle St. Peter founded their church? And, if so, what are the implications with regard to the Roman Catholic Church claiming St. Peter founded their Church in Rome?
There are a few things we must remember prior to going on into details with this question. Prior to 1054, most of Christendom was in Christian unity and the Church in Antioch was no exception.
The Church of Antioch is one of the five patriarchates that constituted the one, holy, catholic, and apostolic Church before the schism between Rome and Antioch in 1098 and between Rome and the other patriarchates at around the same general period. Today it is one of the autocephalous Orthodox Churches. In English translations of official documents, the Church of Antioch refers to itself as the Greek Orthodox Patriarchate of Antioch and All the East.
The Church of Antioch is the continuation of the Christian community founded in Antioch by the Apostles Peter (who served as its first bishop) and Paul, who are its patron saints. In terms of hierarchical order of precedence, it currently ranks third among the world's Orthodox churches, behind Constantinople and Alexandria.
The seat of the patriarchate was formerly Antioch (Antakya), in what is now Turkey. - Church of Antioch (Orthodox Wiki)
As far as the Catholic Church is concerned the Church in Antioch was indeed founded by the Prince of the Apostles St. Peter. Churches can not always control the fates of history and seeing that the Church of Antioch eventually became an independent Orthodox Church, separated from Rome is of interest here. Seeing that the Apostle St. Peter eventually moved on to found the See of Rome where he eventually was martyred for the faith and where the primacy of the Apostle Peter was to remain. In other words, this implies that the gifts that the Apostle St. Peter received from Our Lord were transferred to the successors of St. Peter at Rome and not to the successors of St. Peter at Antioch. That much has never seriously been questioned within the Catholic Church. St. Evodius and St. Peter had different regions of jurisdiction at the same time. The gifts given to the Prince of the Apostles will be transferred to the Successor of St. Peter were he last reigned as Supreme Pontiff.
It is quite possible that St. Ignatius of Antioch, the successor of St. Evodious, was the active bishop of this region prior to the death of the glorious Apostle St. Peter while he was yet the Bishop of Rome.
Now let us look a little closer to the question of Antioch. Scriptures point out to us that St. Peter was indeed in Rome.
The incident at Antioch was an Apostolic Age dispute between the Apostles Paul and Peter which occurred in the city of Antioch around the middle of the first century. The primary source for the incident is Paul's Epistle to the Galatians 2:11–2:14. Since Ferdinand Christian Baur, scholars have found evidence of conflict among the leaders of Early Christianity; for example James D. G. Dunn proposes that Peter was a "bridge-man" between the opposing views of Paul and James the brother of Jesus.1 The final outcome of the incident remains uncertain, resulting in several Christian views of the Old Covenant to this day. - Incident at Antioch
The Church of St. Peter in Antioch, or Antakya, may very well be the world’s oldest place of Christian worship still in active use. Built into the side of a slope on Mount Starius, it was the seat of the first patriarchate established outside of the Holy Land. According to tradition, it stands on the spot where Peter the Apostle delivered the first sermon in the city, and it was here that the term ‘Christian’ was used for the first time in reference to the followers of Jesus Christ. The Church of St. Peter was rebuilt and expanded on several occasions, but is remarkable for the fact that some of the masonry and artwork has been reliably dated to the 1st century AD, making it the oldest Christian church building still in existence. Unfortunately, because of its remote location in a solidly Muslim region, it receives relatively few Christian visitors. Nevertheless the site is well preserved under the auspices of the Turkish Department of Museum Management, and is open to all who do make the journey. - Church of St. Peter
Church of Saint Peter at near Antakya (Antioch), Turkey
Some people believe that the founding of the Church in Antioch can be traced to the Biblical Acts of the Apostles (11:25-27), which relates that Barnabas travelled to Tarsus to bring Paul the Apostle there. They worked for a year with the nascent Christian community, and their adherents to the faith were called "Christians" for the first time. Christian tradition considers the apostle Peter to be the founder of the Church of Antioch and the first priest of the Christian population established there; the Church of St. Peter is traditionally considered to be at the place where he first preached the Gospel in Antioch.
The oldest surviving parts of the church building date from at least the 4th or 5th century; these include some pieces of floor mosaics, and traces of frescoes on the right side of the altar. The tunnel inside which opens to the mountainside is thought to have served the Christians for evacuation of the church in case of sudden raids and attacks. Water which seeps from the nearby rocks was gathered inside to drink and to use for baptism; flow of this water, which visitors drank and collected to give to those who were ill (believing that it was healing and curative), has lessened as a result of recent earthquakes.
Crusaders of the First Crusade who captured Antakya in 1098 lengthened the church by a few meters and connected it with two arches to the façade, which they constructed. Acting on orders from Pope Pius IX, Capuchin Friars restored the church and rebuilt the façade in 1863; French Emperor Napoleon III contributed to the restoration. The remains to the left of the entrance belong to colonnades which formerly stood in front of the present façade. - Church of Saint Peter (Wikipedia)
The Catholic Encyclopedia has this to say:
The first Bishop of Antioch after St. Peter. Eusebius mentions him thus in his "History": "And Evodius having been established the first [bishop] of the Antiochians, Ignatius flourished at this time" (III, 22). The time referred to is that of Clement of Rome and Trajan, of whom Eusebius has just spoken. Harnack has shown (after discarding an earlier theory of his own) Eusebius possessed a list of the bishops of Antioch which did not give their dates, and that he was obliged to synchronize them roughly with the popes. It seems certain that he took the three episcopal lists of Rome, Alexandria, and Antioch from the "Chronography" which Julius Africanus published in 221. The "Chronicle of Eusebius" is lost; but in Jerome's translation of it we find in three successive years the three entries
•that Peter, having founded the Church of Antioch, is sent to Rome, where he perseveres as bishop for 25 years;
•that Mark, the interpreter of Peter, preaches Christ in Egypt and Alexandria; and
•that Evodius is ordained first Bishop of Antioch. - Evodius (Catholic Encyclopedia)
According to the Apostolic Constitutions Book VII P 46, the claim is that St. Evodius was first bishop, but who was appointed as such by Peter; thus making St. Peter also the founder of the Antiochian See.