There are a number of technical issues with making what you describe precise, for example: what precisely is a supermanifold with boundary? how can you glue/compose bordisms? etc. Many or all of these have been or can be surmounted, and I would suggest looking at the work of Stolz-Teichner and the reference therein to see what sort of things people have tried to do.
In any case there is the fact that (in the smooth category) every $(d|k)$-dimensional supermanifold is (non-canonically!) of the form $\pi E$ for $E$ a rank $k$ vector bundle over a $d$-manifold.
Thus, unless you do something more fancy like work in families, when you pass to bordism classes you will just get the bordism group of $d$-manifolds equipped with rank $k$-vector bundles. This has two names:
$$ \pi_d( MO \wedge BO(k)) = MO_d(BO(k))$$
either as the d-th homotopy group of the smash of the spectrum $MO$ and space $BO(k)$ (which is also another Thom spectrum) or equivalently as the d-th $MO$-homology group of $BO(k)$. So that identifies the bordism group.
I think it is a very interesting question whether the whole Pontryagin-Thom construction can be carried out inside the world of supermanifolds, but that is a different question.