We introduce MERAXES, a new, purpose-built semi-analytic galaxy formation model designed for studying galaxy growth during reionization. MERAXES is the first model of its type to include a temporally and spatially coupled treatment of reionization and is built upon a custom (100 Mpc)3 N-body simulation with high temporal and mass resolution, allowing us to resolve the galaxy and star formation physics relevant to early galaxy formation. Our fiducial model with supernova feedback reproduces the observed optical depth to electron scattering and evolution of the galaxy stellar mass function between z=5 and 7, predicting that a broad range of halo masses contribute to reionization. Using a constant escape fraction and global recombination rate, our model is unable to simultaneously match the observed ionizing emissivity at z≲6. However, the use of an evolving escape fraction of 0.05 — 0.1 at z∼6, increasing towards higher redshift, is able to satisfy these three constraints. We also demonstrate that photoionization suppression of low-mass galaxy formation during reionization has only a small effect on the ionization history of the intergalactic medium. This lack of 'self-regulation' arises due to the already efficient quenching of star formation by supernova feedback. It is only in models with gas supply-limited star formation that reionization feedback is effective at regulating galaxy growth. We similarly find that reionization has only a small effect on the stellar mass function, with no observationally detectable imprint at M*>107.5M⊙. However, patchy reionization has significant effects on individual galaxy masses, with variations of factors of 2 — 3 at z=5 that correlate with environment.