The so-called Oka-Grauert principle states that for any Stein space $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. 
The original reference is 



<cite authors="Hans Grauert" mrnumber="98199" cite="_Math. Ann._ **135** (1958), 263--273">_Hans Grauert_, [**Analytische Faserungen über holomorph-vollständigen Räumen**](http://www.ams.org/mathscinet-getitem?mr=98199), _Math. Ann._ **135** (1958), 263-273.</cite>

As a consequence, every locally free, coherent sheaf $\mathscr{F}$ defined on a contractible subvariety $X$ of $\mathbb{C}^n$ is free.

Of course, if $\mathscr{F}$ is not locally free this is no longer true. For instance, take a closed analytic subvariety $Z \subset X$; then the ideal sheaf $\mathscr{I}_Z \subset \mathscr{O}_X$ is coherent but not free.