(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
As for ordinary sheafification, $(\infty,1)$-sheafification is the left exact left adjoint (∞,1)-functor to the inclusion of (∞,1)-sheaves into (∞,1)-presheaves.
For more details see (∞,1)-category of (∞,1)-sheaves.
An $(n,1)$-presheaf can be $(n,1)$-sheafified by
applying the plus-construction on presheaves $n+1$ times (see Lurie, section 6.5.3)
applying a single “hypercover-plus-construction” (see DHI, theorem 7.6)
(∞,1)-sheafification / ∞-stackification
The iterated plus-construction is mentioned in section 6.5.3 of
The “hypercover-plus construction” is discussed around theorem 7.6 of