More specifically, is it possible to leave a startup, buy the equity available to me through my vested stock options and then sell part of it back to current employees of the same startup?
I don't see a reason why not, but maybe I'm missing something. At the end of the day, once you trigger the stock options, you receive a physical stock certificate that you should be able to do as you please with...
> you receive a physical stock certificate that you should be able to do as you please with...
No. The shares you’ll receive after exercising almost always have transfer restrictions. To sell or otherwise transfer them, you’ll almost always need the company’s permission.
Some companies are okay with it, others aren’t, others decide on a case-by-case basis or have specific requirements (like not increasing the size of the cap table). Read the options agreement and then ask the company, all before exercising (and potentially incurring a taxable gain). If possible, do all of this well before leaving.
(What you’re describing is closer to publicly-traded securities, which you mostly can “do as you please with” in terms of dispensation.)