It looks to me like anyone can set up a post and promise prize money. But what is stopping them from just up and leaving when it comes time to pay out?

Does speedrun.com middleman the money first, or is it more like a black market where you just hope the person is trustworthy?

I think you just have to hope

they have no affiliation with the site so choose carefully who you trust

Yeah SR.C itself has nothing to do with monetary rewards or how they're distributed or anything. Would be kinda nice if the site had some kind of middleman system for these kinds of things, but that would introduce a lot of legal issues and liability for them so I don't see that happening.