Wow, that's quite the trip you're making!
I just read your question to my daughters and their mouths fell open with shock. They couldn't even imagine that you would be allowed into school with no shoes. I think they need a little cultural exchange :-)
Your sons will be able to be barefoot in the pool area of your hotel and inside the water parks (Blizzard Beach and Typhoon Lagoon), but that's about it. In general, shoes are requirement in any public area in the United States. Disney World is no exception. If you click
HERE, and scroll down to the area that says "What to Wear," you'll see that shirts and shoes are required in theme parks at all times. You'll find this to be the case in all the restaurants, shops, and public areas of the hotels as well.