The black mage is actually a race in Final Fantasy 9 instead of a job-like the other Final Fantasy games. Most black mages though are manufactured from the mist from the lifa tree -and are magician dolls mass-produced for war, but Vivi was not one of the evil ones, that were just taught to kill. Also most black mages only live a year but Vivi, I believe lived for 6 years. Vivi was a prototype black mage- and was home-made instead of mass produced. I would think though they could not have kids since they are manufactured ?
When something is just manufactured, it's not supposed to bear children. Just like robots, any day one tells us that a synthetic robot got pregnant and gave birth then we would have a mystery in our hands. This is why I'm surprised hearing about vivi and the children.
It is most likely that Black Mages have a way of naturally reproducing. My reasoning for this is that Vivi had an inevitably unhappy ending, known from the first time you go to the Black Mage Village. Having a life expectancy of only a child, Vivi's children manifest the continuation of his race, that they really are alive and that they truly do have hope.