You'll be able to eat pretty much anywhere at Disney, as long as you let them know you have allergies. Some places have more options than others, and the options can change, so go into it with some flexibility.
Email the special diets people 30 days in advance with your trip dates; they send you a PDF of current suppliers (which do change frequently).
Table service meals/ADRs allow you to note the allergy on the reservation; at buffets, they will walk you through the buffet and tell you what you can have, or if it's a regular sit down, the chef comes out to speak with you.
At a counter service place you will just immediately inform the person taking your order. They are required to get a manager to take your order- you absolutely cannot just "invent" a GF option, or order a sandwich on a gluten free bun - so leave some extra time for that conversation. They are well trained. They will not and cannot speak to you once you say "allergy" until the manager has come by so be patient.
The manager comes out with a binder, containing the ingredients for every single thing on the menu. They alone are allowed to touch your food, prepare it from start to finish, and serve it to you. This does take extra time, so the best thing to do is go early or late, not at rush times for meals. I've waited as few as 5-10 minutes and as long as 45. Go early and you go first - let the order-taker know about your allergy, and then let the rest of your party get their orders in while you're waiting for the manager.
For food court situations, it's a bit different from place to place, but generally you walk up to the area between the counters (where the chefs go into the back) and just tell them you have an allergy and need to speak to the chef. Chef Dan at AoA is very helpful and very talkative, and will talk you through the whole process.
Be sure to visit the allergy kiosk in Animal Kingdom in the morning - they have a printout sheet of the gluten free options available at every eatery in the park. They also have some GF snacks stocked there, and Omission Beer (very good!).
Our best successes:
Flame Tree BBQ
Kouzzina
Pop and AoA food courts
Columbia Harbour House
Be Our Guest - good food and GF lemon/raspberry cream puffs!
Casey's Corner
Biergarten
Aloha Isle
Turkey legs were OK when I went, but always check
Boardwalk Bakery had GF sandwiches and Babycakes cupcakes in to go boxes, I'd go on the early end to snag the sandwiches as there weren't very many of them.
Boma - lots of options and the chef was very helpful.
Rose and Crown did an awesome GF fish and chips.
Not so hot:
ABC Comissary (the 45 minute wait for a dry burger, ugh)
Ohana
Kona Cafe didn't have any GF options for tonga toast, major bummer
Cosmic Ray's did their best, but it took forever because of the sheer amount of people. If you go, go when it's not crowded.
I would avoid 'Ohana, as the meal wasn't as interesting without the noodles or the special bread pudding. Just very expensive for what you'll end up getting.
good luck! Disney is a great vacation for us simply because we don't have to stress about the food. I think you'll be pleasantly surprised.