I agree with the food diary option if you are unable to get tested by an allergist. Getting tested is probably the best option as the Allergist will likely put a grid on your back and perform a skin test by placing an allergen in contact and monitoring that site for a reaction. That type of test is the fastest but there are also blood tests and food challenges if needed. It will also let you prepare for reactions by getting an epi-pen prescription if needed in addition to carrying Benadryl.
Not all of my son's allergies were diagnosed at once but we were aware of his reactions and were able to monitor for it and pay attention to what he ate to then follow up with the doctor. For example, we knew my son had dairy and nut allergies but one day he had a severe reaction to something dairy and nut free. He was eating Tahini for snack and he swelled up and exhibited other symptoms such as rashes and a scratchy throat. We took him in and told the doctor what he reacted to and he was tested for sesame which came back positive.
Also there are certain numbers associated with allergies and they can be high or low. My son's numbers for dairy and peanut were very high so we have not done any food challenge for those but his almond numbers were low so we performed a food challenge in the doctors office and he passed for almonds. We still treat him as allergic to tree nuts in general but if it came down to it that almond was the only ingredient, we would now know it's ok for him.
This will be our first trip to Disney with my son and since he has known allergies, when I booked dining, I was able to mark Dairy, Peanut, Tree Nut and Other under the special requests section and then I called in to have Sesame specified as what the allergy under Other is referencing. This will ensure that they are aware in advance as to the types of allergies my son has but we will still ask to speak to the chef personally when we arrive.
I'll know more from experience in November but everything I read is that they are very accommodating to food allergies. It is important to us that not only he be able to eat something but that he'd be able to have choices. We rarely travel because we would have to prep his food somehow with no guarantee we'd find someplace to prepare foods safe from his allergens either as an ingredient or through cross-contamination. The few times we have traveled, it's been typical that whenever we mention my son's food allergies, the servers just assume we mean Gluten-free and we have to explain in detail my son's allergies. In the end, maybe he'll be able to have pasta and sauce or maybe grilled chicken so the prospect of chicken nuggets and fries and other options at Disney will make him very happy.
I've done a lot of research on Disney and food allergies and I still don't know everything and will probably need to look at ingredient books everywhere we stop. We are looking forward to this trip and hoping it will go smoothly with no reaction but the most important starting point for us is that we know what he is allergic to now and can take the appropriate precautions.
Good luck.