I did some research, and apparently the IRS recognizes the national or state 4-H organization under §501(c)3, but the individual chapters—like yours—are incorporated separately and benefit from the charitable status without having to file all the paperwork. So you're getting the best of both worlds: support from a major nonprofit, which handles the legal heavy lifting with regard to charitable status, and significant autonomy within the local chapter (which is incorporated independently as a non-profit).
If mentors donate things to the non-profit (money or gifts), it's probably tax-deductible for them in some form. That has the dual effect of giving the mentors a break on costs, and establishing explicitly that the non-profit owns the stuff (no question of whether they're borrowing it).
I also found some more comprehensive advice about dissolution of non-profits:
here and
here. They confirm my suspicion that upon dissolution, the non-profit's assets will be taxed if distributed among team members. (To avoid diversion of assets with an "exempt purpose".)
Also have a look at the
Nonprofit Good Practice Guide, especially
this.