As long as you are traveling within the U.S. children under 18 do not need any form of ID to fly. If they are traveling on some kind of youth fare, which would be rare, then the airline may require proof that they qualify,
Yep, no ID at all. I just flew with my DD's 11 and 14, both look older than they are. No issues at security at all. At most, they were asked their name and age.