Hi Liz,
As a (sadly, former) vegetarian, I did spend part of that time as a vegan, which included a visit to WDW. While there, I had absolutely no problem with a restricted diet. In fact, I found all the restaurants incredibly helpful. Oftentimes the chef would even come out to speak with me and would make items that weren't even on the menu.
When you make a table-service reservation, make sure to include a notation about not having any animal products. Most often your server will confirm this; but if not, don't hesitate to remind him/her. By the way, my favorite restaurant for vegan choices if
Boma at the Animal Kingdom Lodge.
When you are dining at counter-service restaurants, you can request to see a list of ingredients in each of the food selections.
More information on
special dietary requests can be found in the included link. Within this link are a list of counter service restaurants that can accommodate common food restrictions or already have items on the menu. You may also want to visit www.disneyfoodblog.com and do a search for
vegan for more information and experiences from guests.
And you do need to know that at Downtown Disney there is a baby cakes NYC for some delicious vegan treats!