The case when $M$ is a smooth manifold follows from the smooth Oka principle. See there for an expository account of the argument and references to additional sources.
Indeed, the left side of (*) is $$\def\Hom{\mathop{\rm Hom}} π_0(ʃ\Hom(M,X)),$$ whereas the right side of (*) is $$π_0(\Hom(ʃM,ʃX)),$$ where $ʃ$ denotes the shape functor, which is called “geometric realization” in the main post and is denoted by $‖{-}‖$ there. (From my point of view, a geometric realization functor converts a categorical object like a simplicial set to a geometric object like a topological space, whereas a shape functor converts a geometric object like a sheaf of simplicial sets on manifolds to a categorical object like a simplicial set.)
Concerning the case of a general $M$, not much can be expected if $M$ has homotopy groups in degree 1 or higher (meaning $M$ is not weakly equivalent to a sheaf of sets on manifolds). For example, taking $M=N/\!/G$ to be the stacky quotient of a manifold by an action of a Lie group, and $X$ to be the stack given by the homotopy group completion of the sheaf of symmetric monoidal groupoids of vector bundles, the left side of (*) computes the equivariant K-theory of $M$ (basically, it boils down to Segal's model), whereas the right side computes the Borel equivariant K-theory of $M=N/\!/G$. These two are different in general.