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We present a generalization of the transfer Hamiltonian method to include inelastic currents due to electron-vibration coupling. This formalism is applied to CO vibrations on Cu(100). The method describes the changes in conductance across the vibrational threshold of both the inelastic (increase) and elastic (decrease) contributions, which are evaluated from electronic structure calculations of tip and sample. The most active modes are found to be the two frustrated rotations of the CO molecule. A comparison with previous results, based on the local density of states, shows that explicitly including the tip structure significantly improves the quantitative agreement with experiments. Finally, we discuss how different tip terminations affect the inelastic tunneling current. ; G.T. has been funded by EPRSC Grant No. EP/C541898/1. G.T. and N.L. also acknowledge support from the European Science Foundation, ESF exchange grant, Ref. 1280. M.P. and A.A. have been partially funded by the Spanish MEC, the UPV/EHU, and the Basque Government through the ETORTEK program. W.A.H. is supported by the Royal Society. ; Peer reviewed