Yes, every holomorphic vector bundle of any rank is trivial on the punctured disk $\dot{\Delta}$ . Indeed, since $\dot{\Delta}$ is a Stein manifold ( like any non-compact Riemann surface ! ) the Oka meta-principle (here a Theorem of Grauert ) says that the classification of holomorphic vector bundles on that manifold is the same as that of topological vector bundles. But since the punctured disk is homotopically equivalent to a circle , all topological *complex* vector bundles are trivial, hence the triviality of all holomorphic vector bundles.
(Do not confuse with the Möbius vector bundle, which is a non-trivial  *real* vector bundle!)

**Bibliography** If you want to read complete proofs, you can consult Otto Forster's *Lectures on Riemann Surfaces*.          
On page 229, Theorem 30.3 states that every holomorphic  line bundle  on a non-compact  Riemann surface $X$ is trivial, and immediately below (on the same page) Theorem 30.4 proves that holomorphic vector bundles of any rank on $X$  are also trivial.