I find my kids get hungry at Disney because of all the walking, being off schedule, and on the go-go-go! When the kids were around 4 we had them start carrying their own fanny pack with a small water bottle, sunglasses, and snacks. They're older now, so they are on to a backpack/purse, but the same concept.
I pack them crackers, individual packs of olives, pretzels, individual peanut butter, fruit cups, granola bar, etc. I don't care if they don't eat at the restaurants, so long as they are eating throughout the day and drinking enough water. For snacks - we'll get fruit and cheese plates (Land pavilion has a good one) and powerade. We do usually get one ice cream or a churro a day - because it's vacation!
By carrying their own snacks, they just pull something out when they are hungry. The first time we did it - it was a game changer for me and my sister! No more bugging us for every snack in sight!