Have you emailed Brenda Bennett yet? You'll want to do that.
Every park has one counter-service that sort of "specializes" in the allergy stuff. I do know they have gluten-free hot dog buns. Which my son loved, so that was about all he was interested in the whole week.

I think some can do gf chicken nuggets. But you'll want to head to that counter-service, and don't go during the lunch rush.
As far as table service, in AK we ate at Yak & Yeti and I can't say I'd recommend it for anyone concerned about food allergies. Sooo... since AK has limited table service options... I'd have to recommend Tusker House by default. Never been there, but we wouldn't go back to Yak.

I made a rhyme!
When you contact Brenda, if you tell her your dining reservations, she'll put you in "the system" and that will make it much easier. When you arrive at the restaurant, it should pull up on their computer that there are allergies, if not tell them. They're very good about it. If it's a buffet the chef will walk you thru (I just went ahead and loaded up the plate as we walked), if not then the chef comes to the table and talks thru some options. Most everyone had some sort of gf dinner rolls or bread, Tony's had gf pasta.
If I had to pick one that did the best, I'd say Chef Mickey's, although Tony's was a close second. Chef Mickey's rocked- the chef made gluten-free dairy-free egg-free waffles for DS, they looked home-made not the frozen Van's ones, can't beat that.