Just about every sit down restaurant on property has a some vegan items, so do many of the counter service restaurants. Honestly, counter service meals are the only ones you need worry about in the slightest. Their are fruit and vegetable stands in most of the parks as well for snacks. Right across from Haunted Mansion and the River Boat at MK, HS has one right across from the Beauty and the Beast show and of course EPCOT has Sunshine seasons, Africa and plenty of booths offer vegan snacks during the festivals.
Tangerine Cafe and Sunshine Seasons at EPCOT both have very good counter service selection for everyone.
Animal Kingdom Lodge counter service The Mara, sit down Sanaa, Boma and Jiko all have great food and plenty of vegan options.
Skipper Canteen and BOG both have good food and vegan options in the MK.
The Wave, Cali Grill at Contemporary both have very good food and vegan options.
Satuli, Yak and Yeti, Tuskerhouse and Tiffins all in AK are very good and have vegan options. Highly recommend Satuli for counter service meals. High quality food for everyone. Tuskerhouse IMO is the 2nd best buffet and best Character meal bang for the buck in WDW.
Brown Derby and Brown Derby Lounge is very good at HS. Mama Melrose will have some nice dish's as well.
Just about every restaurant in DIsney Springs has offerings as well.