I think you'll be fine and not have much trouble; generally Vanderbilt students are socially progressive/don't have a problem with sexuality. The gay community isn't very big in Nashville, although there is some presence. Of course there's the chance you'll encounter the occasional rude person, but I think that's almost everywhere. You'd probably want to avoid the redneck bars, but I think that would also apply to anyone who's not a redneck.
In my experience, the closed-mindedness was more about economic status. Many of the students have never had to pay for a thing in their lives, and the sense of entitlement was kind of shocking. Being poor and/or ugly is a much bigger strike than any social issue. LOL I have some great memories from the place, but I also met some of the most childish people ever. This was undergrad, though, and I'd assume the Grad programs (especially SLP) would be a better calibre of students.