I didn't do this for Disney but did it for Universal.
We flew our two nephews from Maine to FL without an adult. We had their parents write up a letter saying that we could make medical decisions for them if the parents could not be reached - followed by the numbers to try the parents at. We did not have their insurance cards, but that wouldn't be a bad idea. If you're flying domestically, you won't need anything on the plane for them. If you are flying from one country to another, then that's different.