Let $V$ be an affine variety over an algebraically closed field $k$ and
$D \subset V$ a Cartier divisor which is normal and has an isolated singularity at $p \in D$.

Let $\mathcal{O}_V^*, \mathcal{O}_D^*$ be the sheaves of invertible functions on $V$ and $D$.
Then I think that we have an exact sequence $0 \rightarrow K \rightarrow \mathcal{O}_V^* \rightarrow \mathcal{O}_D^* \rightarrow 0$.

**Question (Edited)** Is there an affine open neighbourhood of $p \in V' \subset V$ such that $H^0(V', \mathcal{O}_{V'}^*) \rightarrow H^0(D', \mathcal{O}_{D'}^*)$ is surjective where $D':= D \cap V'$? That is, can we lift a surjection of stalks to that on some open neighbourhood?

I think the Question is reduced to the following.

**Question'** Is the cokernel of $H^0(V, \mathcal{O}_{V}^*) \rightarrow H^0(D, \mathcal{O}_{D}^*)$ finitely generated as an abelian group?