<strike>To the best of my knowledge, this has never been "officially" described in the set theoretic literature.</strike> This has been described by Blass and Scedrov in *Freyd's models for the independence of the axiom of choice* (Mem. Amer. Math. Soc. 79, 1989). (It is of course implicit and sometimes explicit in the topoi literature, for example Mac Lane and Moerdijk do a fair bit of the translation in *Sheaves in Geometry and Logic*.) There are certainly a handful of set theorists that are well aware of the generalization and its potential, but I've only seen a few instances of crossover. In my humble opinion, the lack of such crossovers is a serious problem (for both parties). To be fair, there are some important obstructions beyond the obvious linguistic differences. Foremost is the fact that classical set theory is very much a classical theory, which means that the double-negation topology on a site is, to a certain extent, the only one that makes sense for use classical set theory. On the other hand, although very important, the double-negation topology is not often a focal point in topos theory.

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Thanks to the comments by Joel Hamkins, it appears that there is an even more serious obstruction. In view of the main results of Grigorieff in
*Intermediate submodels and generic extensions in set theory*, Ann. Math. (2) 101 (1975), it  looks like the forcing posets are, up to equivalence, precisely the small sites that preserve the axiom of choice in generic extension.