A theory of spectral properties of electron transport via a localized state coupled with a vibrational degree of freedom is presented using a two-channel Newns-Anderson Hamiltonian with a nonequilibrium electron distribution. Our model can be applied to a study of electronic transport through an atomic wire, conductance via a single molecule bridge sandwiched between two electrodes, and also inelastic tunneling spectroscopy of single adsorbates with scanning tunneling microscope. A common key feature expected in these phenomena is an inelastic scattering with lattice or molecular vibration. The density of states ρa of the localized level and the second derivative of the total current d2I/dV2 with respect to the bias voltage are calculated in order to elucidate how the inelastic scattering manifests itself in ρa and d2I/dV2. It is found that two different time scales associated with the lifetime of tunneling electrons in the localized state and the residence time due to virtual excitation of electrons between two electrodes and the localized state play important roles in various features of d2I/dV2 spectra.