We present a calculation of the full set of next-to-next-to-leading-order QED corrections to unpolarized Møller scattering. This encompasses photonic, leptonic, and nonperturbative hadronic corrections and includes electron mass effects as well as hard photon radiation. The corresponding matrix elements are implemented in the Monte Carlo framework mcmule allowing for the computation of fully differential observables. As a first application we show results tailored to the kinematics and detector design of the PRad II experiment where a high-precision theory prediction for Møller scattering is required to achieve the targeted precision. We observe that the corrections become essential to reliably calculate the corresponding differential distributions especially in regions where the leading-order contribution is absent.