Our fam is gluten free, dairy free, and we try to minimize or avoid soy, but no specific allergies to soy.
The Special Diets folks are a huge help, so definitely contact them.
As for each park, we love certain places in each park because we know we can count on them for a good meal with no issues for our son (the one with the specific intolerances). No one wants to think their child might be sick while on vacay

. I do prepare b'fast for my son at home and bring it to Disney with us in a cooler. Gluten free pancakes (frozen) that I can thaw while there and simply warm in the food court while my husband gets b'fast for everyone else, gluten free oatmeal to warm and serve him, etc. It's really not a hassle, and I freeze/prepare in b'fast sized portions so I just pull one out and take it with us to the food court every morning. I also make him almond milk and bring it, but we do get him rice milk at the resort food court from time to time.
In MK, we love Cosmic Rays. The chicken with veggie sides is really good, and enough to easily share. We also like The Plaza Inn for a good Rice Dream treat.
In AK, the fabulous special diets kiosk is WONDERFUL

. We stop and load up on a few snacks and treats. That way, when we stop for a snack elsewhere, I can always reach into the backpack and pull out something for my son. He adores the Rocky Mountain caramel popcorn!
Also, Flame Tree BBQ is almost all gluten/dairy free, so we have no worries about eating lunch there and we do it every.single.trip.
In Epcot, lunch always happens at Sunshine Seasons. Rotisserie chicken, simple veggie sides, and a few allergy-friendly dessert options. Easy!
In DHS, we usually do ABC Commissary, mainly because it's indoors, ha! But they do gf/df chicken nuggets, which my son loves and doesn't get a lot of. Snacks are a bit tough in DHS, unless you like fresh fruit (which my son eats TONS of, so we let him have snacks that are a bit less healthful while in WDW). So this is why I load up on treats/snacks for him--I know I can produce something he'll enjoy even if we're in a place that has few or no allergy-free options.
Honestly, a worry-free trip to WDW is easily do-able, and we've had no problems at all doing Disney allergy-free for years. Best wishes to you!