Proceed to other venues. In organizations without specific programs, you may find someone more willing to work with a preteen or young teenager. Both will take almost anyone. Also, this is a perfect time to think about family volunteering, that is, creating a project on which you and your children work together. After observing your preteen or teen in action, they may make an exception to their age rules.
Preparation is key.
Decide on a field of interest, list organizations devoted to that interest in your community, and write a resume. Suggest that your teenagers approach organizations without volunteer programs with a telephone call. Always mention the connection during the initial phone call. Our daughter Tamara called twenty some-odd vets before she found one that would talk to her and give her a trial run. Maybe the local TV stations are not interested, but the local radio stations will welcome any help they can get.