By and large, this is from my encounters with the EO and how they have come to define it. I have tried to restate, as best as possible, what they've told me. They claim that Sola Scriptura is not valid, or even unbiblical they say, because the Church came first, not the bible. With this, the EO I've talked with say that Jesus didn't write anything to guide the Church, but rather chose apostles to establish and carry out the Church. Thus, Apostolic Succession is the Apostles passing down the Oral tradition to others, and in turn, those pass it down to others. It is true that the Apostles wrote, but using 2 Thessalonians 2:15 the EO say that there are oral teachings that have been passed down (then they obviously say that those are only found in their Church). From my talks with them, they appeal to Apostolic Succession primarily for showing the historical side of the EO. When I bring up the Oriental Orthodox churches, they say they aren't "the One" because they split from the EO after the Council of Chalcedon. Similarly, when I bring up the Great Schism the EO say that the West, RCC, split from them, not the other way around. The EO say it was in the making, particularly with the West attributing more authority to the Pope and the filoque clause added in the Nicean Creed. Thus the EO see the RCC as valid in regards to Apostolic Succession, but is nullified since they haven't held to the correct teachings (they've have added doctrine such as purgatory and penance, etc.). The main thing I get from my encounters with the EO over Apostolic Succession and how they define it is to show that it set up their tradition (that they got it from the Apostles themselves), is what makes the Church infallible, and that it gives authority to their church to be the "interpreters of the Bible" (they stress this because of Sola Scriptura). Because the Bible is the "product" of the Church, it is duty of the Church (EO) to interpret it, I've been told.
I think there's more to the EO claim than what I'm seeing here. To be honest, I do not have a lot of time to invest on the board. I think Wayne has already recommended to you an excellent book by Thomas Smyth which you can download for free. I'll offer a few pointers...
1) The apostolic church itself was never (and this is important to stress) without a
functioning canon of Holy Scripture. I would encourage you not to speak of the Bible
per se because in the early apostolic church there were no codices, i.e., any book form of Holy Scripture. Nonetheless, the apostolic church itself was never without a
functioning canon of Holy Scripture (Acts 17:2, 11; 18:24, 28; Rom 1:2; 15:4; 16:26; 1 Cor 15:3-4; 2 Tim 3:15-16). Acts 17:2 speak of Paul's "custom" in the planting of churches was to reason with Jews and Greeks from the Scriptures.
2) According to Irenaeus, their claim to extrabiblical tradition is no different from the Gnostics...
Irenaeus (c. 130-c. 200): When, however, they are confuted from the Scriptures, they turn round and accuse these same Scriptures, as if they were not correct, nor of authority, and [assert] that they are ambiguous, and that the truth cannot be extracted from them by those who are ignorant of tradition. For [they allege] that the truth was not delivered by means of written documents, but viva voce: ANF: Vol. I, Against Heresies, Book 3:2:1.
Theodoret testified to the same...
Theodoret of Cyrrhus (393-466) commenting on 1 Tim. 6:20-21: Give a wide berth to those hollow profanities and contradictions in what is falsely called knowledge; by professing it some have strayed from the faith (vv.20-21). Those who followed in the steps of Simon called themselves Gnostics; what the divine Scripture is silent on (they claim) God revealed to them, but they are full of impiety and licentiousness. He was right to call this knowledge what is falsely called knowledge: the darkness of ignorance is what they have, not the light of the knowledge of God. Robert Charles Hill, Theodoret of Cyrus: Commentary on the Letters of St. Paul, Vol. 2 (Brookline: Holy Cross Orthodox Press, 2001), p. 230.
3) If you are going to engage these people seriously, you're going to have to invest in some good, apologetic reading materials. Again, Wayne has recommended to you one excellent work. Moreover, if you want to engage them on their own ground intelligently, then you're going to have to become extensively familiar with patristic literature. I do not really recommend this for the average layman because it requires years of reading, study and research. This requires years of study, which is a far cry from what one can possibly gain from a forum like the Puritan Board.
4) Given their misrepresentation of even apostolic history, I would question their sincerity with respect to truth. When one sees how Romanism and Eastern Orthodoxy belittles the importance of the corpus of Holy Scripture, one can scarcely take them seriously when they begin to make claims about the early church, because the early church did not share their shameful view of the same.
5) When they say (according to you) that "Apostolic Succession is the Apostles passing down the Oral tradition to others, and in turn, those pass it down to others," they are making a claim for what "apostolic succession"
does, and that does not define what "apostolic succession"
is according to whatever the presupposition is upon which they're basing their claim for what it does. Claiming that "apostolic succession" gives them this tradition is still no definition of what it is. So, I'm not sure you understand what they mean by "apostolic succession."
I am sorry I do not have more time to invest with you, but I do not think you have articulated clearly what "apostolic succession" means in the way they are using it, at least not in what I have seen of your posts here.