A friend with food sensitivities passed on this information to me. There are several grocery stores in the Orlando area which will deliver to the Disney resorts.
This is a huge relief for me. Although Disney has promised to accommodate Alex's food requirements, he's still likely to need some supplies for private snacks and meals. I had planned to do a taxi run to a local grocery store (and had tracked down 3 within a 10 minute drive of our resort) but if we can have the food delivered, that would be awesome.
We can stock up on the basics of milk, bread, peanut butter, cheese and some fruit to keep in the room (which comes with a fridge). Then meals and snacks can be readily available and brought with us. Including for the flight home, which will be a difficult day.
I've gone through a couple of websites and it looks fairly straightforward. Check off what you want and indicate the day you want it delivered. Pay via a credit card and the front desk will receive it, letting you know once its arrived.