Hi! We are vegans and vegetarians (I'm vegan) and have eaten at WDW several times over the past few years.
Our favorite place is Boma at AKL because it's an African buffet with a LOT of interesting vegan options. The chef will walk you through the line, and show you what is vegan, and they are knowledgeable about things like gelatin in desserts. There aren't a lot of vegan desserts, but they will bring you vegan ice cream or sorbert with cookies and berries. Sanaa is amazing as well.
AK: Tusker House is called "Boma Light." There's also a food cart with falafels and steamed edamame pods in AK. There are also fresh fruit carts, frozen pineapple skewers, real fruit popsicles, and large pickles
Ep: Tangierine Cafe and Sunshine Seasons (quick service), Via Napoli can do delicious cheeseless pizzas, kaki gori shaved ice in Japan. Don't try eating in France.
HS: not much (Hollywood and Vine has some ops, as does Mama Melrose, but nothing spectacular).
MK: Crystal Palace buffet has some vegan ops. Columbia Harbor House has vegan chili and steamed broccoli. Also, Dole Whips are vegan, as are all soft pretzels and popcorn in the parks
Contemporary: 1900 PF isn't as amazing without Chef TJ, but they do make a delicious stir-fry for vegans, plus there is veggie sushi and a few salad and veggie dishes at the buffet you can eat, and they will prepare you a vegan dessert
Poly: Ohana will make a tofu and veggie stir fry
Beach Club: Beaches and Cream has a veggie burger and lots of vegan ice creams for dessert
A lot of food courts have make your own pasta or salad with toppings like chickpeas, olives, and so forth
Try checking out menus here:
http://allears.net/dining/menu/
And recommendations here:
http://allears.net/din/vegtsparks.htm
I'm sure I've forgotten some. Have fun. Disney is a great place to eat vegan.