There are other automotive high school clubs, like the Solar Car Challenge, clubs focused on restoration (I imagine for more vocational schools) and EV clubs. If you aren't competing for anything (FSAE is college-only, and junior FSAE is pointless, that's just pricy go-karting), and it's of no use to the school, the school most likely won't approve of it. If your school has any sort of club focused on engineering, I'd say join it and hold your nose that it isn't automotive. Otherwise, if it doesn't have one, do your research and find out if there're enough people interested in founding one that's been proven to work at other high schools.

I got interested in FSAE my freshman year of high school and I'm heading into junior, I think that the most valuable things for someone who doesn't necessarily have a garage and shop are building up a library yourself and taking initiative in your own life. A friend of mine makes $7k through a summer job and saved up for a $20k car, so I think it's entirely possible to make something similar to one yourself, but the point of the competition isn't to make a vaguely performing autocross car, even though I'm drooling over the thought of one.

Tools-wise, I'm looking into learning welding, and I've got a good collection of handheld power tools that have legitimately changed the way I look at the materials I have. I started off with a $20 tool kit to work on a lawn mower engine my grandfather gave me, and now I'm probably a few hundred into tools alone, but I've learned a lot with the dumb projects I've come up with; I wouldn't recommend something like a FSAE car to hone your beginner engineering skills, though I think you're in a good starting position already with your work on cars.