Damian Rössler pointed out an essential obstruction which is in fact the only one: a holomorphic vector bundle $E$ is the pullback of the universal (quotient) bundle on a Grassmanian via a holomorphic map if and only if $E$ is generated by a finite number of holomorphic global sections. The point is that unlike the $C^\infty$ case, a holomorphic bundle need not have any nonzero global sections at all.