Electronic transport in a model molecular device coupled to local phonon modes is theoretically analyzed. The method allows for obtaining an accurate approximation of the system's quantum state irrespective of the electron and phonon energy scales. Nonlinear electrical features emerge from the calculated current-voltage characteristics. The quantum corrections with respect to the adiabatic limit characterize the transport scenario, and the polaronic reduction of the effective device-lead coupling plays a fundamental role in the unusual electrical features.