I have been with vegan friends before, but it has been a few months. So the suggestions below were what they were able to eat at the time, but Disney does change things around a lot. So I would definitely check before ordering.
DCA:
Pacific Wharf Cafe: Vallejo Vegetarian Chili
Oatmeal for breakfastl
Paradise Garden Grill: Mediterranean Skewers with tofu and veggies, but only a couple of the sauces were vegan, so check! Of course, the yogurt sauce is not, but there was another that wasn't vegan either. This was one of my friends' favorite meals at DCA for sure. We ate here twice.
Lucky Fortune Cookery: Asian rice bowl with tofu and veggies, ask about the sauces again, they do have a vegan one.
Fidler, Fifer, and Practical: Starbucks coffee with soy!
Cocina Cucamonga: ended up with a LOT of substitutions here, but ended up with the Soft Tacos Monterey with veggies, no cheese, no side of sour cream, and steamed veggies on the side with no beans or rice. There was a bit of a communication problem, so I don't know for sure that the beans/rice aren't vegan, so double check.
Wine Country Trattoria: pasta dish with broccolini, sorry my notes are vague on this one. But there was a vegan option!
Disneyland:
Jolly Holiday: Grilled veggie and whole grain salad. My notes says basil vinaigrette was very good.
Coffee with soy milk!
River Belle Terrace: Becky Thatcher Fruit Plate, generous portions of fruit.
Royal Street Verandah: Vegetarian gumbo (I have no idea if this has changed, but it was one of the best vegan options that my friends liked while were at Disneyland, definitely worth asking if it is still vegan.)
Carnation Cafe: Chef's Vegan Burger
Rancho Del Zocalo: Veggie Fajita Style tacos or burrito, they also said they could make the tostado salad without cheese and it would be vegan. Obviously, no cheese and no sour cream. Here, they were very certain the beans were vegan, but there was some disagreement regarding the rice. No problem, they gave us double beans.
Blue Bayou: didn't eat here, but were told the Portabello Mushroom and Couscous Maque Choux was vegan. Obviously, double check.
Cafe Orleans: Vegetable Ragout without the corn cakes OR the cheese on top. Substituted with side salad with no problem.
Mickey Mouse shaped pretzels at the stands, no cheese obviously.
Plaza Inn Breakfast with Minnie: Chef came out and said the gluten free Mickey waffles could be made dairy free and would be vegan. This was fine with the fake syrup (lol) and the tons of fresh fruit available.
Bengal BBQ: the veggie skewers themselves are vegan, but cooked on a grill with meat, so fyi.
---There are lots of fruit stands throughout both parks that sell many different kinds of fresh fruit. These were very helpful as snacks.
---Definitely double check, but we were told that the pre-packaged kettle corn sold at the candy stores is vegan. Didn't try it.
---We also had the steel cut oatmeal for breakfast from the Jamba Juice in Downtown Disney. They make it with soy milk and have tons of fruit options. Also, Jamba Juice makes some great vegan smoothies that really helped in the afternoon.
Hope you have a great time!