Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$ with isolated zeros.

Does there exist a sequence of vector fields $V_n \in C^\infty \cap W^{1,2}$ on $\mathbb{R}^2$, such that $V_n \to V$ in $W^{1,2}$ and the $V_n$ do not vanish on $\mathbb{D}^2$?

If we replace the $W^{1,2}$ convergence with $L^2$ convergence, than the answer is positive. The idea is to push the zeroes out of the disk by composing $V$ with a diffeomorphism which affects a region of very small measure.

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.

Does there exist a sequence of vector fields $V_n \in C^\infty \cap W^{1,2}$ on $\mathbb{R}^2$, such that $V_n \to V$ in $W^{1,2}$ and the $V_n$ do not vanish on $\mathbb{D}^2$?

If we replace the $W^{1,2}$ convergence with $L^2$ convergence, than the answer is positive. The idea is to push the zeroes out of the disk by composing $V$ with a diffeomorphism which affects a region of very small measure.

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$ with isolated zeros.

Does there exist a sequence of vector fields $V_n \in C^\infty \cap W^{2,2}$ on $\mathbb{R}^2$, such that $V_n \to V$ in $W^{2,2}$ and $V_n$ do not vanish on $\mathbb{D}^2$?

If we replace the $W^{2,2}$ convergence with $L^2$ convergence, than the answer is positive. The idea is to push the zeroes out of the disk by composing $V$ with a diffeomorphism which affects a region of very small measure.

I am interested to know whether stronger convergence is possible.

This question arose in the context of this answer, where the existence of such an approximation is assumed.

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$ with isolated zeros.

Does there exist a sequence of vector fields $V_n \in C^\infty \cap W^{1,2}$ on $\mathbb{R}^2$, such that $V_n \to V$ in $W^{1,2}$ and the $V_n$ do not vanish on $\mathbb{D}^2$?

If we replace the $W^{1,2}$ convergence with $L^2$ convergence, than the answer is positive. The idea is to push the zeroes out of the disk by composing $V$ with a diffeomorphism which affects a region of very small measure.