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The answer is NO!
For example, let $X=S\setminus\{s\}$ be the complex manifold obtained from a compact Riemann surface $S$ of genus $\geq 1$ by removing an arbitrary point $s\in S$ .
Then every non-empty open subset $U\subset X$ has non denumerable Picard group $\operatorname {Pic}(U)=H^1(U, \mathcal O^*)$ , which proves that the acyclic cover you require does not exist.
(By the way, note that $X$, like all non compact Riemann surfaces, is Stein)