I had long heard that a lathe was the first thing you should get, and a mill was optional. Given the non-round nature of the things I was working on, I was skeptical. I got a benchtop lathe and mill, and my skepticism seemed justified: I barely used the lathe, and spent all my time on the mill.
Now that I have a full-sized mill and lathe, I mostly use the lathe, and the mill really does seem optional.
The point here is that it all depends on the equipment: the benchtop lathe I got was unusable out of the box, and the benchtop mill worked right off the bat. That is an important thing to keep in mind: these benchtop lathes are widely considered to be a kit rather than a get-it-and-go piece of machinery like a drill press. I would definitely say that the cheap import benchtop lathes should
not be anyone's first lathe.
But you shouldn't get discouraged. Take a couple weeks to scout out what used machinery goes for in your area (craigs list, etc). Maybe even talk to the sellers of the ones you can't quite afford - they might give you a break when they hear about your project. Just say "here is what I am using it on, do you think I could do that" and then if (when) they say yes, ask if they would accept 500 in cash for it.
In the meantime, work on the instructors who can oversee the shop area. Some of them might be teaching other classes over the summer and could be convinced to do their grading in the shop while you use the machines. If there is no overlap with the summer staff, the same instructors might know some of the summer staff who would do this. It is key to talk to the teachers themselves: the administrators are going to be worried about insurance and liability and things costing more money, but the teachers are going to be more interested in helping a motivated student accomplish a significant and worthwhile goal, and they will then work with the administrators to get things squared away. And remember, if you pull this off, the
school comes out looking good - they'll be showing pictures of your project to parents for decades, saying "look what we taught one of our sophmore students to do!" (yes they will take all the credit, and none of us here will get any).
I don't think leasing machines is even possible for low-end stuff, so I won't propose that. But a corrollary to the "working on the instructors" angle is to borrow tooling from the school shop so that you don't have to buy it. Of course if you break it you have to replace it, so be careful there, but they might furnish you with stuff like safety equipment, faceplates, clamps, tool holders and the like. Really depends on the instructor, it would have to be under the table as no administrator would go near the idea
EDIT: oh, and don't forget the GoFundMe route. You're not asking for a lot cash, maybe 300-500 to make the project feasible.