We report the development of a vibrational-electronic convergent close-coupling method for electron-molecule scattering with an ab-initio account of the coupling between the electronic and vibrational motions. The technique has been applied to scattering on molecular hydrogen, including coupling between vibrational levels in the first 11 electronic states. Distinct resonances associated with the temporary formation of the H_{2}^{-} ion are present between 10 and 14 eV for numerous transitions, including vibrational excitation of the X ^{1}Σ_{g}^{+} state, dissociation via the b ^{3}Σ_{u}^{+} state, and excitation of the B ^{1}Σ_{u}^{+} state. With both resonant and nonresonant scattering treated in a single calculation, this method is capable of providing self-consistent sets of cross sections for electron-molecule scattering in regions where the adiabatic-nuclei approximation breaks down.