What class of elements are providing the many electrons needed to make the H- ion?
Easily ionised ones like sodium and potassium. Not much from lithium because that is rare.

You don't need many free electrons. The fraction of H$^{-}$/H is very small, something like $10^{-7}$ in the solar photosphere. And indeed the number densities of sodium and potassium to hydrogen are of that order.