Basically, this issue comes down to whether the team is a separate legal entity (e.g. an incorporated non-profit under 26 U.S. Code §501©) or is legally considered part of something else (e.g. part of the 4-H organization, the school board, or a partnership among team members).
If your team is legally separate, then the property and debt belong to the corporation. However—and this is the big reason why corporations exist—the officers of the corporation have limited liability with regard to debts and contracts. (The big exception being that criminal activity carries full liability.) As FIRST teams go, there are a few that have gone the route of registering under §501©3. As I understand it, they usually do this for the autonomy it grants them, because it means that they can isolate their finances and day-to-day operations from the school board. If you want to dissolve one of these, you’ll need to plan for dealing with any debts before passing out the assets to team members. And there might well be tax implications if a team member receives some asset from the corporation.
If you’re part of the school board, you can probably take advantage of their §501© status. But depending on your particular situation, and especially if you use their funds to pay for your expenses, all of your stuff probably belongs to them. So if you dissolve the team, the board is supposed to inherit it all. (Don’t go giving out stuff that isn’t yours!)
If the team is basically an informal club, then you have few restrictions. But there’s one huge catch—the officers of the club bear full liability for pretty much everything. Forget to pay for something? The creditor could file a lawsuit, and the officers of the club will personally end up as respondents (rather than only if negligence is alleged, as is the case with a corporation).
I’m not familiar with 4-H’s organizational structure, so I don’t know which of those scenarios best approximates what you’ve got. Hopefully you can clarify—would you be able to give some more details of your associations with 4-H (e.g. at national/state/county levels), the school boards and the responsible adults?
There’s actually another scenario that has come up with FIRST teams, but which probably doesn’t apply to you: the sponsor thinks that because they give the team money to build robots, that they own the robot (and/or other assets). I’m of the opinion that this is largely without merit, absent a contract that states this—there’s no expectation of tangible return when you make a donation, so the sponsor is not entitled to anything. It doesn’t hurt to make this distinction explicit in your bylaws—basically distinguish a donation (free money) from a contract (money in exchange for goods or services).
Please note that I can only speak in generalities, because my experience in these issues is mainly with organizations within Canadian public universities, and your situation will surely differ. In particular, there’s a whole set of rules governing American non-profits that impose stringent obligations and restrictions on the scope of your activities. If you want to go the route of becoming a standalone non-profit, you’re going to want to consult with a lawyer—on a pro bono basis, hopefully. (This post is not intended to be anything but a starting point for your research.)
