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. 



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> 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 $\mathbb{D}^2$?
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If we replace the $W^{1,2}$ convergence with $L^2$ convergence, than the answer [is positive](https://math.stackexchange.com/a/3439203/104576). 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.



Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Going from the topological index idea: let $C$ be a circle around one of the isolated zeros, and let $D$ be a disk containing $C$. By the trace theorem, your vector fields in $W^{1,2}(D)$ restricts to vector fields $W^{1/2,2}(C)$ on the circle. Since they do not vanish, you can regard them as vector fields $W^{1/2,2}(C, \mathbb{S}^1)$. (If there's any problem with my argument, it would be this step, passing from an $\mathbb{R}^2$ valued function to $\mathbb{S}^1$ valued one.)


The $W^{1/2,2}(C,\mathbb{S}^1)$ functions are continuously embedded in $VMO(C,\mathbb{S}^1)$ since $C$ is one dimensional, and one can use the $VMO$-degree theory (originally due to Boutet de Monvel and Gabber, and extended by Brezis and Nirenberg, see [this survey by Brezis](https://link.springer.com/chapter/10.1007/0-8176-4467-9_3)). The upshot is that for continuous functions the VMO degree coincides with the the usual topological degree, and the $VMO$-degree is continuous under $VMO$-convergence. So I think [Pietro's argument against the $W^{2,2}$ case using degree theory](https://mathoverflow.net/questions/346238/can-we-approximate-a-vector-field-on-the-plane-with-non-vanishing-vector-fields#comment866546_346238) should carry over also, telling you that you shouldn't be able to do your approximation. 


To be more precise: the chain of arguments should go something like: since your $V\_n$ are in $C^\infty$ and has no zeros, their corresponding topological degree would be zero, measured either in $VMO$ or classically; but $V\_n$ converges to $V$ in $VMO$ (by argument above), and this means that $V$ has to also have degree zero, a contradiction. 


