What class of elements are providing the many electrons needed to make the H- ion? Continuous absorption by the H- ion reaches its maximum in the cool atmospheric layers of G stars.  At these temperatures, most of the hydrogen atoms are neutral; but what class of elements is 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.