In general, you are allowed to bring food with you around WDW. We have also previously been advised to talk to special dining when we needed food accommodations. Here is the email:
Special.Diets@DisneyWorld.com
If he eats PB&J, then I can tell you WL almost certainly carries Uncrustables.
My experience is more with food allergies, but I have also visited WDW with someone with autism, and a limited diet.
If I may be so bold, the above person was often described as a picky eater. But picky makes it sound like he was choosing to be difficult. He wasn't. He was almost 10 when we was finally tested for food allergies. Turns out, he had numerous mild food allergies to many common foods. Food allergies vary in intensity. A mild food allergy can appear to be a simple dislike. Several foods just made his mouth itch, or he just felt a little off when he ate them, but as a child, he did have the words to express that.
He also has/had sensory difficulties. He outgrew many of them, but when he was little...the smell of garlic was like nails on a chalkboard to him. He once ran out of an Italian restaurant absolutely SCREAMING with his fingers shoved up his nose and tears streaming down his face. When he finally calmed down, and sat to eat, the bread they gave us was full of rosemary. So that again caused an overload. Luckily they also stocked plain bread.
In any event, if they carry a food ANYWHERE at Wilderness Lodge, in a situation like yours, they can usually serve that food in any dining location in that hotel. Advance notice helps. They can't get foods they don't carry, but they want small children to be happy. If they know you are coming, they will try to have that food on hand. Of course I can't promise they will, but talking to them is the first step.
One thing they that cannot do is reheat or otherwise prepare outside food. I think, if possible, they would rather supply foods to you. That's what we have been told.
On top of that, food allergies and food intolerances run in my extended family. Few people understand food intolerances. An intolerance = symptoms, but not the same ones as a true allergy. The person who has one did not know until they were in their mid-40's. Doctors first thought the symptoms were lupus, but a smart doctor thought to check for a food intolerance. Turns out wheat was causing a broad range of symptoms: sinus symptoms, acne, stomach pain, muscle aches, rashes, etc. They all went away when he stopped eating wheat.
Prior to that, he might have been described as a picky eater. When he ate foods like pizza with bell peppers, he burped pepper taste. So he blamed peppers as the trigger, when it was the pizza crust.
Getting tested was a game changer for both of them, but it took some investigation.