My DS is now 5 and has been PA since he was 18 months old. We've been to WDW many, many times and have never had a problem. We've eaten at most all of the restaurants and most all of the kid's meals are already peanut free. We did skip the Coral Reef when they had the peanut butter playdough dessert (wasn't about to even try and get the tables clean).
You used to HAVE to speak to the Chef about the food, but I'm finding more and more that the servers can help you now. Of course, you are dealing with many more allergies that I have to. Not one time did any Chef or server give us any kind of attitude when we were talking with them.
Chef Mickey's was the only place that we had any trouble with and it was only the dessert. Just like someone else posted, the only dessert safe was the ice cream. We got lucky in that the server brought us some "fresh" sprinkles from the back and DS was fine. We're lucky that vanilla ice cream with sprinkles just happens to be one of his favorites!
Oh, and just in case you happen to have a reaction, just go to any Cast Member. I'm sure you'll have your own Epi-Pen, but in Florida the paramedics are allowed to carry epinephrine. If you tell them it's an anaphylactic food allergy, they will hurry. If you tell them it is a CHILD with an anaphylactic food allergy reaction, they will hurry even faster!
Have a great trip!!