I'm lactose intollerant and I've always found things to eat at Disney. If you are at a sit down restaurant within Disney World just ask to speak to the chef regarding your son's allergy. They will let you know the options that your son can have and usually will offer to make something entirely off the menu if necessary. The chefs are all very nice and most willing to accommodate dietary restrictions.
For counter service I just usually eat things that I am 99.9% sure are dairy free. I stay away from anything batter dipped, breads, dressings, desserts, etc.
On the snack carts they serve fruit bars which are dairy free - those are always a nice treat!
I know that your son will have plenty to eat. I always do!! Have Fun!!