Studies of topological semimetals have revealed spectacular transport phenomena spanning extreme magnetoresistance effects and ultrahigh mobilities. As phonon dynamics and electron-phonon scattering play a critical role in the electrical and thermal transport, we pursue a fundamental understanding of these effects in type-I Weyl semimetals NbAs and TaAs. In the temperature-dependent Raman spectra of NbAs we reveal a previously unreported Fano line shape, a signature stemming from the electron-phonon interaction. Additionally, the temperature dependence of the A1 phonon linewidths in both NbAs and TaAs strongly deviate from the standard model of anharmonic decay. To capture the mechanisms responsible for the observed Fano asymmetry and the atypical phonon linewidth, we present first-principles calculations of the phonon self-energy correction due to the electron-phonon interaction. Through this study of the phonon dynamics and electron-phonon interaction in these Weyl semimetals, we consider specific microscopic pathways that could contribute to the nature of their macroscopic transport.