Do the Sobolev inequalities as they are stated on Wikipedia also hold on bounded open subsets of $\mathbb R^d$ whose boundary is not $C^1$ but Lipschitz?
If so, what is a good reference for a proof?
Actually, I'm looking for a reference where the inequalities are proved for any bounded, open set which has the "extension property" (see, for example, [Renardy, Definition 7.11]).
Yes, those inequalities still hold. You only need the construction of the extension operator, which can be found, for example, in Stein's Singular Integrals.