Puffballs live on dead organic matter.
They are usually produced in soils with a large amount of black, amorphous organic matter like lawns and pastures. I assume that it might take years for a mycelium to get large enough to produce the large puffballs. I do know from research reports that only 1 in a million puffball spores germinates. So large soil cores transplanted to a new pasture rich in organic matter might work. If I were trying to isolate the fungus in pure culture, I would probably try transplanting a bit of tissue from the inside of a young puffball to get a pure culture.