Hey there, Paula. Thank you so much for your questions.
You are absolutely welcome to bring food into any of the Walt Disney World theme parks, as long as there aren't any glass bottles or alcoholic beverages. Whatever you need to bring for your son, you can definitely bring with you.
In the
Baby Care Centers, located at each of the theme parks, you'll find a microwave oven that you can use to warm things up if that makes your life a little easier. You can also plug in your warmer in those locations if you decide that's a better way to go. You may have to explain to the security guard what the device is and how you need to avoid cross contamination with other foods, which is what makes the warmer necessary.
There are different size lockers available at the four theme parks. The jumbo lockers, available at Magic Kingdom and Epcot, are probably the ones you are going to have the most luck with. They are 17" x 22" x 26". The large lockers, available at all four parks, are 15.5" x 13" x 17".
I also want to say that Walt Disney World is one of the most incredible places when it comes to
special dietary needs. In many cases, you can meet with a member of the culinary team and have them make something geared specifically for your son. I know it's a scary proposition to put your son't special diet in the hands of others, but it's definitely an option to consider. My good friend has celiac disease, and just came back from a very successful trip through the Disney restaurants!
Happy planning!
Anna