This is obviously a question of state law. Most state nonprofit corporation laws have an age requirement for individuals who can serve as directors of a nonprofit corporation, but not for those who can serve as members.  This makes a certain amount of sense since members have no fiduciary duty to the organization and can act in their own self-interest.  Directors have fiduciary duty and can be sanctioned if they don't fulfill it.

Adults may be perceived to have more experience and judgment to act on behalf of others than younger persons do. But where the members have all the power and authority, when permissible under the law, they have no fiduciary duty to anyone else and there is no necessary reason to preclude the kids from voting.  Officers would have fiduciary duty, but not the broad group of members who make the policy decisions for the organization.

This structure may not be a good idea (either having no directors, or having kid members), but it is probably not illegal.