Nobody needs ID to board the plane; just a boarding pass.
Travelers 18 years of age and older need a government-issued photo ID to get through the TSAsecurity checkpoint. ID for these travelers is occassionally needed for secondary checks at the gate, but these checks also would be conducted by TSA not by the airline at boarding.
Your children will not need any kind of ID unless they are traveling on some kind of youth fare. In that case, you may need to provide proof of age (birth certificate) at the airline check-in desk outside the security checkpoint.