Does the std::stack in the C++ STL expose any iterators of the underlying container or should I use that container directly?
Stack does not have iterators, by definition of stack. If you need stack with iterators, you'll need to implement it yourself on top of other container (std::list, std::vector, etc). Stack doc is here.
P.S. According to a comment i got from Iraimbilanja, std::stack by default uses std::deque for implementation.