Disney is amazing with allergies. Last time we went 2 years ago, I was allergic to peanuts, shellfish, and eggs.
During that trip, we ate at 50s prime time, crystal palace, sunshine season, cosmic rays, pop century food court, flame tree BBQ, and a few other places I can't remember. Outstanding service everywhere.
We are leaving for Disney in 19 days and in addition to peanuts, eggs and shellfish, I am also now allergic to wheat, oats, coconut, broccoli, tree nuts, and sesame. In addition I am very sensitive to soy and dairy so I try to avoid food with those. I also have pretty severe Oral allergy syndrome to most raw fruits and veggies. So it's complicated.
I've eaten at
Disneyland with these allergies and they are wonderful. From my experiences, Disney is great about cross contamination issues.
I can give you an update when we get back but right now, we are planning on eating at:
Pop century food court
Crystal palace
The plaza
Whispering canyon cafe
Sci Fi dine in
50s prime time
1900 park fare
Liberty tree tavern
Yachtsman steakhouse
Cosmic rays starlight cafe
Sunshine seasons
I would expect that gluten, soy and dairy would definitely be manageable for Disney especially since they are all top 8 allergens and fairly common so I imagine that they encounter these particular allergies a lot.
But I know there are individuals on these boards with those same allergies and are very helpful so hopefully they will be able to give you some information.