Look for the university archives. They are generally open to the public and, in most university libraries, are the ones who run the rare book/manuscript rooms--I've worked at 3 archives through college and law school (one state school, one Seven Sisters, and one small Midwestern school) and that is the way it was handled at all three of them.
In the Yale directories, you'd be looking for Bob Smith, University Archivist, or Jane Smith, Special Collections Librarian, to give you some idea what to look for. I can't speak to Yale in particular, though.