In my opinion, Plato's views on knowledge shifted around in a number of important ways over his career, but you're right to say that he always treats knowledge as having at least some innate aspects. In Book V of the Republic, for example, he characterizes knowledge as a kind of power that is innately within us, but--as a power, rather than a state--this account also recognizes the possibility of developing that power, or allowing it to wither. Much of the discussion of Books VI and VII of the Republic concern how to develop this power--how to "empower it," as it were) through the right educational curricula and with the use of what Plato calls "summoners" (by which he means things that summon the exercise of this power, rather than the inferior cognitive powers of belief or sense perception. There is, accordingly, a social aspect (via education) even in Plato's account.
When philosophers these days talk about "social epistemology," it can mean various things. But a "social theory of knowledge" would seem to imply that knowledge is, by its very nature, a social phenomenon. The place to look for this sort of account (if I have understood you rightly) in the ancient world would be in the writings of the group known as the Sophists, some of whom seemed to think that knowledge was a (purely) social construction. I am not aware of too many contemporary philosophers who accept such a view, but I hope my answer has helped at least a little!