Does a boundary condition undermine a Sobolev embedding result?
I assume imposing a boundary condition on the functions in the space $W^{m,p}(\Omega)$ does not change its embedding into any larger space; for any changes on the boundary, the function still belong to that same class of functions.
Imposing boundary conditions means you are considering subspaces of $W^{m,p}(\Omega)$, thus all the embedding results transfer. You only have to make sure that you end up with closed subspaces, so that the new space is again a Banach space.