A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective)
was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper
“Assembly maps in bordism-type theories”,
especially Sections 3 and 6.