We just returned and have a daughter, not with a peanut allergy, but an egg allergy. Hopefully I can shed a little light for you!
As far as table service- we did all buffets for dinners and at every one the chef made her an ice cream with sprinkles, and/or Divvies cookies. Now, mind you, egg MAY be more prevalent in baked desserts than peanut... but being I have no experience with peanut allergy, maybe I'm wrong. They WILL be able to tell you exactly what dessert on the buffet your child can and can't have if peanut is in some but not all.
In quick service there are always frozen yogurt or Mickey ice cream bar/ice cream sandwich. My daughter thoroughly looked forward to her daily Mickey Bar.
In quick service meals, they were very helpful- but they are strict about "made in a facility that also uses... etc" I didn't push them on it, but if you said you weren't that worried, they MIGHT go with you on it. Not sure.
All this being said, if you e mail about your allergy, they will e mail you back with 2 packets of info on things in the park and resorts that your son will find to eat, and also they will send you a form to fill out with your reservations on them- they will contact every chef and each chef will have a copy of that form. I happened to write on the comments section that my daughter esp. liked grilled meats, rice, mashed potato, steamed brocolli and green beans- and at one of the buffet meals the chef literally came out "Oh, is this the one that loves grilled meats and steamed broccoli?" He made her a plate special of each thing I named. Really above and beyond.....
The email is for the Special Diets group.... you can find it online when you make your dining reservations and note an allergy...
Good luck! OH and other desserts I saw at quick service were - rice krispy treats, magic bars, brownies, cookies, cupcakes.... so you are definitely going to need to be steered to what they can have!