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)