I don't expect to much of a black market during the first phase since the vaccines are being sent directly to hospitals and nursing homes. Medication in those places tends to be literally locked down (initially just for drugs of abuse but recently I've heard most medication is becoming tracked and locked down). Although since the vaccine has to be stored in a freezer, I'm not quite sure how hospitals will keep their supply secure other than having a guard.

Once the second phase opens next year and allows anyone with a risk factor including some pretty common ones; diabetes and obesity, I expect the rules to be bent a bit more and a black/greymarket to gain traction. Although, wealthy people tend to be older anyway which makes them more likely to qualifying risk factors.