We were at WDW during this past Christmas with my daughter and son-in-law. She's vegetarian and he is strictly vegan. Almost all the WDW restaurants can be veg/vegan friendly. Some of the restaurants are outstanding both for what they offer and for excellent guest service.
Among their favorites were ~~
Rose & Crown in England ~~ Had PS for dinner (7:30-got an outside table with a spectacular view of Illuminations). We stopped by the restaurant at noon to make sure they had 'vegan' noted on the PS. The day chef came out and asked my SIL to return later that day to speak with the evening chef. When we returned, the chef came out and spent at least 15 minutes with my SIL talking with him about what he could do for his meal. His meal that evening was perfect.
Boma at the AKL ~~ Excellent choices here. One of the chefs walked my DD & SIL through all the food pods to point out everything they could eat. This restaurant continues to be one of our favorites.
Spoodles at the BW ~~ Here the server was very knowledgeable about veg/vegan choices, but he also had the chef come to the table to make sure everything would be just what they wanted. Good choices here.
Jiko at AKL ~~ Very good veg/vegan items, but our dinner here was marred by a very unpleasant server. Would return here for the food, but would definitely not sit in this server's area again.
Mama Melrose at MGM ~~ Best choices for vegans at MGM.
Beaches and Cream at BC ~~ This was one of the favorites

I think they ate there at least 3 times over the course of the week.
Columbia Harbor House at MK ~~ Good vegan chili here.
I would agree with the other poster about getting the book mentioned. It has very helpful info and lists many more restaurants, both fast food and "nice" restaurants. There are other places we had wanted to try, but they'll have to wait until next trip. One other note ~ do make PS's and have them note "vegan" on the PS; call the restaurant the previous day or early the day of to make sure that's on there.
