Reaction of HOD+ with CO was studied over the collision energy (E-col) range between 0.18 and 2.87 eV, for HOD+ in its ground state and with one quantum in each of its vibrational modes: (001) - predominantly OH stretch; (010) - bend, and (100) - predominately OD stretch. In addition to integral cross sections, product recoil velocity distributions were also measured for each initial condition. The dominant reactions are near-thermoneutral proton and deuteron transfer, generating HCO+ and DCO+ product ions by a predominantly direct mechanism. The HCO+ and DCO+ channels occur with a combined efficiency of 76% for ground state HOD+ at our lowest E-col, increasing to 94% for E-col around 0.33 eV, then falling at high E-col to similar to 40%. The HCO+ and DCO+ channels have a complicated dependence on the HOD+ vibrational state. Excitation of the OH or OD stretch modes enhances H+ or D+ transfer, respectively, and inhibits D+ or H+ transfer. Bend excitation preferentially enhances H+ transfer, with no effect on D+ transfer. There is no coupling of energy initially in any HOD+ vibrational mode to recoil velocity of either of the product ions, even at low E-col where vibrational excitation doubles or triples the energy available to products. The results suggest that transfer of H or D atoms is enhanced if the atom in question has a high vibrational velocity, and that this effect offsets what is otherwise a general inhibition of reactivity by added energy. HOCO+ + D and DOCO+ + H products are also observed, but as minor channels despite being barrierless and exoergic. These channels appear to be complex mediated at low E-col, essentially vanish at intermediate E-col, then reappear with a direct reaction mechanism at high E-col.