I don't think any of my picks are vegan but I enjoy many of the octo/lacto vegetarian items in WDW.
Columbia Harbor House in the MK has my favorite veggie sandwich-the Lighthouse-and their veggie chili is pretty good too. Tortuga Tavern has a vegetarian burrito and a toppings bar but it always seem to be closed whenever I visit.
At Epcot I love the veggie plate at Tangierine Cafe and the roasted beet and goat cheese salad at Sunshine Seasons.
I recently tried the veggie sandwich at Restaurantosaurus in the Animal Kingdom and it was fine, tasty but not as good as the one over at the MK. Restaurantosaurus also has a toppings bar. The fruit plate at Flametree BBQ in the AK is fairly decent too.
Wolfgang Puck Express in Downtown Disney has several great veggie options. I've enjoyed nearly everything I've ever ordered here :-)
I tend to order fish at the table service places but do enjoy the vegetarian options buffet at Boma in the Animal Kingdom Lodge. Also at AKL is Sanaa. I haven't tried it yet but it looks wonderful! I also like the risotto dish at Raglan Road in Downtown Disney.
Edited: 2:34 pm, December 17, 2012