Here is the first thing you need to do.
Email these people:
WDW.Special.Diets@disney.com and ask them for their special diets packet.
They will send you a digital folder with information on common dietary restrictions and where to find the food to fit that.
There is also a form for you to fill out (notating your name, ressie #, the type of allergy, and where you are eating). Fill it out and send it back at least 2 WEEKS BEFORE YOU ARRIVE.
Fill it out with the ressie numbers for your TS meals. They will stamp your seat ticket with an allergy/special diet and a chef is to come to your table to speak with the diet-restricted individual to find out what they can fix for them. Always double check with the server and make sure you speak to a chef before you order.
For CS places, just ask at the counter to speak to a chef. They will grab someone to talk to you about your options.
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I have a black pepper allergy, which, if you ask the chefs, is just as bad as a salt-restricted diet considering salt and pepper are staple seasonings. I had a lot of trouble with the buffet at Chef Mickey's. The only thing I could have on the bar was salad (no dressing...) and dessert. None of the other items on the buffet were okay for me to eat.
Each chef at each place I ate (with the exception of Restaurantosaurus) made a special plate for me. I didn't have to worry about it at AK because the meal I wanted didn't have pepper at all, they looked it up!
Disney is exceptional at accommodating dietary needs. It may take a few minutes to get your order squared away, but I didn't have to worry about a meal the entire trip.
All they will have to do is ask to see a chef at each CS place. They should be able to find something they can eat at each place.
I will recommend going to Sunshine Seasons at EP. He will have A TON of options there.