It's not appropriate for us to help you install proprietary software here.
I recommend you complain to anyone who has any responsibility for requiring you to use Flash Player. I had the exact same problem when I took a college class. My professor was evasive and uncooperative, but you'll never know unless you try. Demand to be allowed to do your work without proprietary software, and explain the issue. Be sure to be respectful, of course; you don't want to sound like a lunatic. I also suggest you use the word "libre" (either on its own or in the combination "free/libre"), so they are not confused.
If you are able, threaten to choose another instructer and talk to the others about the issue. Part of the problem with my ability to convince my professor to let me just do my homework on paper was I was already well into the class.
Also, avoid compromises. If installing Flash would be an ethical compromise for you (which I'm assuming is the case, though I don't know just how much), make this clear. Otherwise, they might consider the problem "solved" because you have the ability to use one of their computers.
I hope you are able to convince someone responsible to let you do the class without proprietary software. If you are unable to, use the school's resources, not volunteer resources, to get help with fulfilling their ridiculous requirement.
