I was just there a week ago, and have a severe gluten allergy.
If you go to City Hall first thing, they can give you a menu specific to your child's allergy. I got one for gluten-free food, and it lists ALL of the food in Disneyland and California Adventure that I'm able to eat.
It did not list Downtown Disney restaurants, but they are all super knowledgeable and accommodating about allergies - they take them very seriously. (In fact, at the Rainforest Cafe, ONLY a manager can take an allergy food order, and ONLY the manager can deliver it to your table to make sure that it was done correctly.)
I felt sort of embarrassed at first, but do not hesitate to ask questions of the managers at Downtown Disney, or the chefs in the parks. They really are wonderful, and WANT to work with you to make the meal the very best that it can be.
Really, no worries at all. But make sure that you get the peanut-safe food list from City Hall! That was awesome.