The question of ownership has not come up where I am. We assume the school owns everything unless there is clear evidence to the contrary. When I bring a tool into the lab, I take a pic and write it off my taxes as a donation. If I do not want to donate it, I print out a big P-Touch label that says “Personal Property of Mr. Allen.”
Parents and sponsors donate, and the team uses the stuff.
Our teams (two FLL, one FRC) are organized as a 4-H club on a college campus, though our funds (and thus, the ownership of the team materials) go through the University Foundation. You may want to consider organizing under some structure or another if it’s not a few friendly parents working together. 4-H clubs, scouts, church groups, homeschooling associations, or just a plain ol’ non-profit are all common not-a-school teams in FLL.
As for the cost, it does start to rack up. Our two teams ran up a bill with LEGO and FIRST into the low thousands between the kits and registrations, and then you have items like our field table. But if you’re game to spring for it, by all means!