We went to Disney World last April/May, the first time since my sister had been diagnosed Celiac.
We ate table service at Kona Cafe (Polynesian) and Jiko's (Animal Kingdom Lodge). At Kona, the chef came out and talked use through the menu, pointing out which items she could have. At Jiko's, our server went through it with us. In each case, a couple of the items she could eat, some she could eat with some changes (changing what it was served with, for instance) and for some he offered to make special gluten-free versions (her own special sauce, for instance). From memory there were only 2 or 3 main course which she could not have. Obviously a lot of the desserts were out of the question but she did have options.
Overall they were very accommodating and couldn't do enough to help us. Unlike many restaurants outside of Disney World, they did actually seem to know what they were talking about and they clearly deal with it a lot. Just make sure you tell them you are celiac when you make ADRs, when you check in and when you meet your server. However, we didn't have reservations for Kona Cafe, but they were just as accomodating as Jiko's.
As for counter service, we did find that at Pizza Planet at the Hollywood Studios, they serve gluten-free pizzas (they are not on the menu, they have to ask). They are cooked especially and so take quite a while to cook. Tip - make sure they don't give you the pizzas for the rest of your party until the GF pizza is ready, otherwise they'll all go cold while you're waiting. The GF pizza is not great, but the consolation is that neither are the regular pizzas!
We ate a few times at Cosmic Ray's at MK - there are enough 'plain' options there (e.g. roast chicken, mashed potatoes and vegetables) that shouldn't be a problem for celiacs. We also had a baked potato for lunch from the stall in Liberty Square in MK.
One website we found helpful was
http://allears.net/din/guestceliac.htm.
Overall, you don't have too much to worry about - eating out Gluten Free at Disney is much easier than in the Real World.