Certainly, such a gadget exists. Let $\newcommand{\Pg}{B\mathrm{Pin}_n^{(m)}}\Pg$ be the homotopy fiber of the map $BO_n\to K(\mathbb{Z}/2, m+2)$ classifying the Stiefel-Whitney class. If we realize the homotopy fiber as the path-space, then the map $\Pg\to BO_n$ is a Serre fibration. Apply the Kan loop group construction to get a surjective map of simplicial groups which does what you want, I think. (You don't ask for a central extension, and I don't think my construction gives one; the kernel of the map of simplicial groups is not going to be abelian, though it models $K(\mathbb{Z}/2,m)$.)

As for an explicit construction, there's quite a bit of work on giving explicit constructions of $\mathrm{String}_n$, which sits in a central extension $K(\mathbb{Z},2)\to \mathrm{String}_n \to \mathrm{Spin}_n$. I particularly like Chris Schommer-Pries's paper "Central extensions of smooth $2$-groups and a finite dimensional string $2$-group", which produces an extension of $2$-groups, from which one can extract an extension of topological or simplicial groups.