We stayed at at the Yacht club and would walk over to the Beach Club's Market for breakfast every morning. We didn't want to have a sit down breakfast except for Chef Micky's which was awesome. If I remember correctly, no peanuts or tree nuts were used in the entire buffet, but you should check again.
Our plan was to be there when the park opened and leave when we needed a break so we could nap or swim or just regroup. Just know, especially with younger children you cannot do it all!! Pick and choose what you think the kids want to do most. We let the kids pick what interested them the most and made sure to get those rides/attractions done first.
We had dinner reservations every night except for one and I think that was the hardest night to find dinner. Kids menus had typical stuff...mac & cheese, pizza, etc. was served with veggies and/or fruit. When you make reservations on line, there is a box to check that says special diet and just click and it will give you the opportunity to specify food allergies.
We ate lunch at the parks. Counter service restaurants have a notebook with all the ingredients. When you go up to order, tell the person you have food allergies and you will be asked to wait for the manager.
I carried snacks and we bought water at the parks.
Are there any restaurants in particular that you are interested in?