We investigate the conduction-band structure and electron mobility in rocksalt ScN based on density functional theory. The first-principles band structure allows us to obtain band velocities and effective masses as a function of energy. Electron-phonon scattering is assessed by explicitly computing the -dependent electron-phonon matrix elements, with the inclusion of the long-range electrostatic interaction. The influence of free-carrier screening on the electron transport is assessed using the random-phase approximation. We find a notable enhancement of electron mobility when the carrier concentration exceeds . We calculate the room-temperature electron mobility in ScN to be 587 at low carrier concentrations. When the carrier concentration is increased, the electron mobility starts to decrease significantly around and drops to 240 at . We also explore the influence of strain in (111)- and (100)-oriented ScN films. For (111) films, we find that a 1.0% compressive epitaxial strain increases the in-plane mobility by 72 and the out-of-plane mobility by 50 . For (100) films, a 1.0% compressive epitaxial strain increases the out-of-plane mobility by as much as 172 , but has a weak impact on the in-plane mobility. Our study sheds light on electron transport in ScN at different electron concentrations and shows how strain engineering could increase the electron mobility.