We analyze electroproduction of light vector mesons (V=rho, phi and omega) at small Bjorken-x in the handbag approach in which the process factorizes into general parton distributions and partonic subprocesses. The latter are calculated in the modified perturbative approach where the transverse momenta of the quark and antiquark forming the vector meson are retained and Sudakov suppressions are taken into account. Modeling the generalized parton distributions through double distributions and using simple Gaussian wavefunctions for the vector mesons, we compute the longitudinal cross sections at large photon virtualities. The results are in fair agreement with the findings of recent experiments performed at HERA and HERMES.