If you had a specific $A$ that led to $S$, then no, you cannot recover $A$ from $S$, even if you knew that $A$ is finite ($S$, of course, will be infinite, because there are infinitely many tautologies that will all be in $S$).
For example, if you had $P \land Q \in S$, you don't know whether you had $P \land Q \in A$, or both $P \in A$ and $Q \in A$, or maybe all three, or maybe both $P$ and $P \land Q$, or maybe both $Q$ and $P \land Q$, or maybe just $Q \land P$, or ...
Something you could do, is to find a kind of 'minimal' set whose closure is $S$. See prime implicants for one way to think about that. However, the techniques discussed there are for propositional logic, and for first-order logic things will get a lot more complicated ... indeed, you may well run into the problem of the undecidability of first-order logic when trying to find such a 'minimal' set.
Finally, as I said, $S$ will be of infinite size, so there are some serious practical considerations to deal with here as well!