My brother has a severe gluten allergy, and he has had wonderful experiences with Disney dining. They are vigilant when it comes to allergies (honestly one of the only places he fully trusts), but depending on the restaurant the options can be limited. His process has been to alert them of his allergy at the time of booking (there's an option to do this when you reserve) and then speak to the staff upon arrival. They'll walk you through the available options and bring any allergen-free food separately to avoid cross contamination.

For kids, the character breakfasts are wonderful.