We calculate non-perturbatively the inelastic effects on the conductance through a conjugated molecular wire-metal heterojunction, including realistic electron-phonon coupling. We show that at sub-band-gap energies the current is dominated by quantum coherent transport of virtual polarons through the molecule. In this regime, the tunneling current is strongly increased relative to the case of elastic scattering. It is essential to describe the full quantum coherence of the polaron formation and transport in order to obtain correct physics. Our results are generally applicable to one-dimensional atomic or molecular wires which became semiconducting due to a Peierls distortion.