  Despite considerable efforts, accurate computations of electron-phonon and
carrier transport properties of low-dimensional materials from first principles
have remained elusive. By building on recent advances in the description of
long-range electrostatics, we develop a general approach to the calculation of
electron-phonon couplings in two-dimensional materials. We show that the
nonanalytic behavior of the electron-phonon matrix elements depends on the
Wannier gauge, but that a missing Berry connection restores invariance to
quadrupolar order. We showcase these contributions in a MoS$_2$ monolayer,
calculating intrinsic drift and Hall mobilities with precise Wannier
interpolations. We also find that the contributions of dynamical quadrupoles to
the scattering potential are essential, and that their neglect leads to errors
of 23% and 76% in the room temperature electron and hole Hall mobilities,
respectively.
