When we stayed at POFQ we did pop over to POR. The chefs were NOT aware that we could be coming; Special Diets only notified POFQ, not POR (can't say I blame them; I don't think I mentioned that we'd pop over a few times). So, here's what I know of their ability to do with no preparation.
We didn't go over for breakfast, but every single food court at WDW Resorts I believe can make allergy safe Mickey waffles. I'd just mention to Special Diets in your questionaire that you'll be definitely wanting those.
My kids did get the chicken tendeners, Silk soy milk (available in the drink fridge to grab yourself), milkshake made with Silk & Tofutti ice cream (choice of chocolate and vanilla).
One thing that they surprised me by not carrying is Tofutti cheese slices. With notice they could probably get them in but they didn't seem to have them on hand when we went.
They did carry Ener-G pizza crusts but no cheese. I don't know if these have eggs or not since my kids can't eat them for other reasons. They really are terrible crusts IMO. The chefs at POFQ premade gluten, milk, egg, yeast and nut free pizzas for my girls because I mentioned how much it would mean to them to be able to get that. The chefs were WONDERFUL!!! The chefs at POFQ also prepared a batch of gluten, milk, egg and nut free brownies for my girls (safe for our level of nut free, not sure about yours; they use Bob's Red Mill mix but I believe that's also what they use for the waffles so if you have problems with Bob's Red Mill then you need to discuss that with the chefs).
They also carried Ener-G hamburger buns but again, I don't know if those have eggs or not since my girls can't eat them for other reasons. Boy do I wish Ener-G didn't use citrus fiber in their products. Sigh....
We had one chef at POR offer to try to come up with a ranch-like salad dressing for DD10 but we told him not to bother since it was our last time over at POR that trip. She would have loved it I'm sure. I'm just trying to show you that they will try their best to make something yummy that meets your own tastes.
With advanced notice, chefs can and will do a lot for you. We do tend to eat at least 1 meal per day at our resort so I like to contact chefs at our resort to make sure that there will be options for our kids. If I was only eating 1 or two meals at the resort I probably wouldn't bother because they do carry enough to give a bit of variety for a couple of meals.