Such an inequality cannot exists. Take $\Omega=B_1(0)\subset \mathbb{R}^n$ and assume find a constant $C>0$ independent of $\omega$, then taking a sequence $(w_k)_k \subset L^p(B_1)$ weakly converging to a delta at $0$ you would prove that for every $f\in C^0(B_1)$ there holds $$ \|f-f(0) \|_{L^q(B_1)} \le C \|\nabla f \|_{L^q(B_1)} \,.$$ But this is not true whenever points have zero capacity (e.g. $p=q=2$ and $n\ge 2$), since it would be equivalent to a Poincaré inequality without any condition.

Such an inequality cannot exist. Take $\Omega=B_1(0)\subset \mathbb{R}^n$ and assume find a constant $C>0$ independent of $\omega$, then taking a sequence $(w_k)_k \subset L^p(B_1)$ weakly converging to a delta at $0$ you would prove that for every $f\in C^0(B_1)$ there holds $$ \|f-f(0) \|_{L^q(B_1)} \le C \|\nabla f \|_{L^q(B_1)} \,.$$ But this is not true whenever points have zero capacity (e.g. $p=q=2$ and $n\ge 2$), since it would be equivalent to a Poincaré inequality without any condition.