The WishList Member plugin can be used with most available WordPress forum options (if that forum uses the WordPress User database for users).
WishList Member creates Member accounts using the WordPress User database. So if you use a Forum plugin that works for Users, it would likely work for Members as well.
If you add a forum to a page in WordPress and protect that page using WishList Member, then non-members would not be able to access that page (or the forum on it). Only members assigned to the Level(s) with access to the protected page containing the forum would be able to access the forum.
Some popular WordPress forums are bbPress, Asgaros Forum, wpForo, Simple:Press, etc.