The only separate "vegetarian menus" I've heard of are at San Angel Inn at Epcot, and V&A at the GF. Mostly, there's a vegetarian entree or two on the menu, or you can make meals of appetizers and sides if there isn't. Chefs can make things on the fly at TS restaurants, especially if you give them a heads-up and mention you're a vegetarian when you make your ADRs. CS places are a bit trickier, and not all are suitable for veggie eating (like Casey's in MK).
Buffets and breakfast menus are always easy.