You can't buy cards from the Black Market on a Mission turn. "Can't" wins.
It's not the case that Black Market fails to do anything due to the rules being "you can't buy cards then." First, the rules aren't like that; the rules give you permission to do certain things at certain times, and you can't do other things by virtue of not having been given permission. Second, Black Market would get to overturn such a rule anyway; cards get to contradict rules.
People in general find this very intuitive. Black Market would be meaningless if it didn't actually succeed at letting you buy a card; that would make no sense and so must not be the case. Mission would be meaningless if it didn't actually stop cards from being bought; it must actually work.
Mission could be phrased so as to not have a contradiction. It's a classic issue that I always think of in terms of the interaction in Magic between "Destroy target land" and "Enchanted land can't be destroyed" (Stone Rain / Consecrate Land). You instead say "When enchanted land would be destroyed, it isn't," and now everything (that did not actually confuse people) is clear (as it happens, these days they have a new game-defined term, indestructible, that gets used in this specific case).
However in general those phrasings that get rid of the non-confusion are super-confusing. The two "replacements" that Dominion has (Trader and Possession) are super-confusing and mistakes. So there was no pull towards Mission saying "That turn, when you would be able to buy a card, instead, you can't."