I don't know about apple juice (my kids are the only kids I know who won't drink juice), but all of the counter service and table service places have milk.
Yes you can get Apple juice all over the parks. Most of the little food carts have it or you can get it at any of the Counter Service Meals. I had no problem getting Apple juice for my 21 month old when we went to WDW last month.
If you are travelling in the summer please remember that milk takes a while to digest and if it is hot and you are going on a bunch of rides milk sitting in a tummy curdling may not be what you want coming back up. So the juice may be a better idea for lunch. But everywhere we went had milk-not carts but CS places.I thought the serving size was large so if little guys they may want to share.