[proofplan] This is the Oka-Grauert principle for vector bundles. Injectivity says that a topological isomorphism between holomorphic bundles can be deformed to a holomorphic one; surjectivity says that a topological bundle can be given a holomorphic structure. Both assertions follow from the Oka principle for maps from Stein spaces into complex Lie groups, applied to transition cocycles with values in $\operatorname{GL}_r(\mathbb{C})$. [/proofplan]

[step:Describe bundles by transition cocycles] Choose a Stein open cover $\{U_i\}$ of $X$ over which the relevant rank-$r$ bundles are trivial. A holomorphic vector bundle is represented by a Čech $1$-cocycle \begin{align*} g_{ij}:U_i\cap U_j\to\operatorname{GL}_r(\mathbb{C}) \end{align*} with holomorphic transition maps. A topological complex vector bundle is represented by the same data with continuous transition maps. Isomorphism of bundles corresponds to changing the cocycle by a $0$-cochain with values in $\operatorname{GL}_r(\mathbb{C})$. [/step]

[step:Use the Oka principle for $\operatorname{GL}_r(\mathbb{C})$] The group $\operatorname{GL}_r(\mathbb{C})$ is a complex Lie group and hence an Oka manifold. For Stein source spaces, the Oka principle says that continuous maps and cocycles with values in an Oka manifold can be deformed, with the required compatibility conditions, to holomorphic maps and cocycles. In cohomological language, the natural map from holomorphic nonabelian Čech cohomology to continuous nonabelian Čech cohomology, \begin{align*} H^1(X,\mathcal{O}(\operatorname{GL}_r))\to H^1(X,\mathcal{C}(\operatorname{GL}_r)), \end{align*} is bijective. [/step]

[step:Deduce surjectivity] Given a topological rank-$r$ bundle, choose a continuous transition cocycle. By the Oka principle, it is cohomologous in the continuous category to a holomorphic transition cocycle. That holomorphic cocycle defines a holomorphic vector bundle whose underlying topological bundle is the original one. Hence the forgetful map is surjective. [/step]

[step:Deduce injectivity] Suppose two holomorphic rank-$r$ bundles become isomorphic after forgetting the holomorphic structure. Their holomorphic cocycles are cohomologous by a continuous $0$-cochain. The Oka principle deforms this continuous cochain, with compatibility with the two holomorphic cocycles, to a holomorphic $0$-cochain. Therefore the two holomorphic bundles are holomorphically isomorphic. The forgetful map is injective, completing the proof. [/step]