It also depends on any special restrictions the court may have placed on an individual.
I know of one particular individual, a neighbor I had at one time, who was restricted from possessing any weapons, black powder, archery, knives or other wise. He had been convicted of a federal crime. It involved violent circumstances, so the court issued an order of no weapons of any type to be in his possession.
And also, if a person is convicted and on probation or parole, their supervising officer can issue a no weapons of any type order, to include knives, black powder, or archery.
As for purchasing, anyone can walk into a sporting good store and purchase black powder weapons. Whether it complies with a court ordered restriction or not is another issue. But there isn't any current system, background check or other, that would prevent such a purchase from taking place.
GS