For counter service:
I think just about every place that serves hamburgers also has available a vegetarian burger. Even if you don't see it on the menu, just ask.
Most places that have ceasar salads with chicken, also have a version without. Also tossed salads available in some places.
There are many places with pizza. Mac and cheese will be offered on the kids menu but that doesn't mean you can't order it.
There are veggie sandwiches or wraps available in nearly every CS location.
Table service:
If you check out the menus you will see that there is usually a vegetarian entree on every menu, and of course salads and other types of veggie appetizers.
In a TS location, you also can ask for the chef and they can usually make some additional recommendations.
Even in buffet locations, always check with the chef so you can be sure that selections aren't made with "hidden" items like beef broth. We've also had lots of luck asking for a meat-free pasta dish when the one on the buffet did have meat in it.
WDW is one of the best places for vegetarian diners.
Check out the dining report board above. Lots of great pictures and descriptions provided by other DIS veggie pals.