You will be able to enjoy a vegetarian or vegan meal just about any QS or TS restaurant in WDW. The only exception I found was at Geyser Point at WL (We don't make that kind of food, was their response!!!)
One of my fav items was the Veggie Rice Bowl topped with lettuce, tomato, etc, etc at Pecos Bill. Asking for plain rice makes it a vegan dish. Sooooooo good!!!!
Enjoy!