We explore the transport of fermions through a quantum conductor in the presence of contact vibrations. The latter are coupled to charge transfer between the fermion reservoirs and the conductor but remain inert to the charging state of the conductor itself. We derive explicit expressions for charge transfer rates into and out of the conductor which extend the scope of rate theories of inelastic transport to the presence of contact vibrations. Implementing the theory to a simple model with a uniform vibronic coupling at different contact orbitals, we demonstrate and analyze the effect of such vibrations on the charge current. Asymmetry between contact vibrations at the two reservoirs is shown to induce a pronounced current rectification, especially in the limit of floppy (low frequencies) contacts. At high frequencies, vibrational quantization is shown to suppress the effect, in accord with the increasing contact rigidity. This quantum result requires corrections beyond the classical theory of charge hopping.
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