The sources that reionized the universe are still unknown, but likely candidates are faint but numerous galaxies. In this paper, we present results from running a high-resolution, uniform volume simulation, the Vulcan, to predict the number densities of undetectable, faint galaxies and their escape fractions of ionizing radiation, fesc, during reionization. Our approach combines a high spatial resolution, a realistic treatment of feedback and hydroprocesses, a strict threshold for minimum number of resolution elements per galaxy, and a converged measurement of fesc. We calibrate our physical model using a novel approach to create realistic galaxies at z = 0, so the simulation is predictive at high redshifts. With this approach, we can (1) robustly predict the evolution of the galaxy UV luminosity function at faint magnitudes down to MUV ∼ −15, two magnitudes fainter than observations, and (2) estimate fesc over a large range of galaxy masses based on the detailed stellar and gas distributions in resolved galaxies. We find steep faint end slopes, implying high number densities of faint galaxies, and the dependence of fesc on the UV magnitude of a galaxy, given by the power law: log fesc = (0.51 ± 0.04)MUV + 7.3 ± 0.8, with the faint population having fesc ∼ 35 per cent. Convolving the UV luminosity function with fesc(MUV), we find an ionizing emissivity that is (1) dominated by the faintest galaxies and (2) reionizes the universe at the appropriate rate, consistent with observational constraints of the ionizing emissivity and the optical depth to the decoupling surface τes, without the need for additional sources of ionizing radiation.