I was staying in a house so I had a kitchen but I didn't even really plan on eating in the parks. This may not be an option for you, but I brought a small lunchbox and a tupperware container from home. We had groceries delivered so I made sure I had soy yogurt, hummus, crackers, grape tomatoes and carrot sticks delivered for me. I brought that almost everyday for lunch. It was the best way to ensure I could eat. I have food allergies and don't eat meat so I am a challenge to feed most places. if you can do this it will make things easier for you.
Every counter service has a big binder so you can read ingredients. Be prepared for it to take longer for your food to come. Everything is premade and if you order something special, it will take long. The manager also has to come and see you. It is policy.
At HS the cinnamon apple stuffed pretzel is vegan (or was) and at the Backlot Express I had the veggie sandwich re-made for me with no pesto or cheese. Delicious! The fries are also vegan (or they were). Be sure to read the binder. They just changed the veggie burger from the vegan Boca patty to one with rice and corn etc. I missed my boca burger! They also have Ener-G rolls b/c last time I checked the buns had milk. So read that binder for all components of your meal. I was able to get the veggie burger at the CS near Tower of Terror in HS and in the country US at Epcot.
The Sunshine Season is a great place for vegans to eat. Lost of veggie choices. Read the binders though!
I wanted to get an veggie egg roll at AK but the line was HUGE so I never even made it to read the binder. That may be a tasty lunch option.
The TS restaurants are great, as well as buffets (breakfast options). The chef will come and speak to you and make sure your meal is safe to eat.
Yes, you will have to sign paperwork for icecream, but that is a liability thing for Disney. The chef will have to scoop it for you from the Plaza or the bakery, so be prepared to wait. It's the rules. I also got tofutti with peanut butter sauce at Beaches and cream and I think a dairy free chocolate sauce they had. YUM!
Let me know if you have any other questions. I do remember getting home and being so happy to eat "real" food again. I lost weight in Disney b/c the options are limited. How much hummus can a girl eat? I was desperate for some tofu. Protein can be hard to come by so gobble it up when you can find it!