I'm allergic to soy along with dairy, mustard, alcohol, all types of sea food, tapioca, brazil nuts, gluten, and all artificial sweeteners. This can make eating out a bit of a challenge but I've been able to get something at each park. It's nothing fancy, but it's food and it's safe. If soy is your only allergy, you'll likely find even more options.
MK: I've gotten plain grilled chicken and fruit at CHH, grilled steak and a "salad" at BOG, and a hamburger patty lettuce wrap with cucumber salad at Cosmic Rays.
Epcot: I avoid CS at every county but the US where I've gotten steak and plain steamed broccoli. Sunshine Seasons has grilled chicken, roasted potatoes, and lots of allergy friendly packaged snacks.
Animal Kingdom: This one has been a bit of a challenge but I've still gotten a plain hamburger patty and fruit at dinoland. They had plain meats at the barbecue place. There's also the new allergy kiosk here to stock up on snacks.
DHS: I haven't eaten here in a LONG time because I can get is a plain hamburger again. I usually go to a different park or a resort if I'm spending the day at here.
Resorts: I'e gotten wonderful chicken stir fry at Captain Cooks at the Poly and the Design-Your-Own-Pasta at POR is safe too.
I'll also admit to having a bucket of popcorn for dinner.