We present a first-principles approach for inelastic quantum transport calculations based on maximally localized Wannier functions. Electronic-structure properties are obtained from density-functional theory in a plane-wave basis, and electron-vibration coupling strengths and vibrational properties are determined with density-functional perturbation theory. Vibration-induced inelastic transport properties are calculated with nonequilibrium Green's function techniques; since these are based on a localized orbital representation we use maximally localized Wannier functions. Our formalism is applied first to investigate inelastic transport in a benzene molecular junction connected to monoatomic carbon chains. In this benchmark system the electron-vibration self-energy is calculated either in the self-consistent Born approximation or by lowest-order perturbation theory. It is observed that upward and downward conductance steps occur, which can be understood using multieigenchannel scattering theory and symmetry conditions. In a second example, where the monoatomic carbon chain electrode is replaced with a (3,3) carbon nanotube, we focus on the nonequilibrium vibration populations driven by the conducting electrons using a semiclassical rate equation and highlight and discuss in detail the appearance of vibrational cooling as a function of bias and the importance of matching the vibrational density of states of the conductor and the leads to minimize joule heating and breakdown.