How do I write a resume as a high school student + freelance developer?

There's different formats of resume. So while some people do a chronological listing of all their past positions, you can also use the "Skills Resume" format -- where you make a list that breaks out all the individual areas in which you have skills. (And since you're just interested in one client, you can sneakily tailor this list so it includes all the things they're looking for in this position. Just don't make it too obvious that that's what you're doing.) :)

There should be examples on the web. The only thing I'd add is that resume writing is incredibly common -- millions and millions are written every year -- so there's lots and lots of information out there. (There's even services that will help you write a resume.) But also: your high school probably has a guidance counselor whose main job is to help people apply to college, but who might also be able to advise you about writing a resume. (Or at least refer you to someone who will.)

I guess I'm saying: don't be afraid to ask for help. No one's born knowing how to write a resume. And the one thing I never really realized as a teenager was that there's adults who get a good feeling from just knowing that they helped someone. I can imagine there's people who'd be thrilled at a chance to help a teenager write their very first resume -- so if you want someone to help you, you'd probably be able to find someone willing to offer advice.

Honestly. If the client doesn't want to hire a minor, it's probably for a good reason. You should really be focusing on college applications anyway, Peter Thiel's exhortations notwithstanding.