(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of $sSet$-site is the incarnation of the notion of (∞,1)-site when (∞,1)-categories are incarnated as simplicially enriched categories.
An $sSet$-site is a simplicially enriched category $C$ together with the structure of a site on its homotopy category $Ho(C)$.
This appears as (ToënVezzosi, def. 3.1.1)
Under the identification of simplicially enriched categories with models for (∞,1)-categories, $sSet$-sites correspond to (∞,1)-sites.
Because, as discussed at (∞,1)-site, that is equivalently an (∞,1)-category equipped with the structure of a site on its homotopy category of an (∞,1)-category.
For $C$ an $sSet$-site, the local model structure on sSet-presheaves is a presentation of the (∞,1)-topos $Sh_\infty(C)$ over the (∞,1)-site corresponding to $C$
model site, simplicial site