We study the magnetic field evolution of an isolated spiral galaxy, using isolated Milky Way-mass galaxy formation simulations and a novel prescription for magnetohydrodynamic (MHD) supernova feedback. Our main result is that a galactic dynamo can be seeded and driven by supernova explosions, resulting in magnetic fields whose strength and morphology is consistent with observations. In our model, supernovae supply thermal energy, and a low level magnetic field along with their ejecta. The thermal expansion drives turbulence, which serves a dual role by efficiently mixing the magnetic field into the interstellar medium, and amplifying it by means of turbulent dynamo. The computational prescription for MHD supernova feedback has been implemented within the publicly available ENZO code, and is fully described in this paper. This improves upon ENZOs existing modules for hydrodynamic feedback from stars and active galaxies. We find that the field attains $mu G$-levels over Gyr-time scales throughout the disk. The field also develops large-scale structure, which appears to be correlated with the disks spiral arm density structure. We find that seeding of the galactic dynamo by supernova ejecta predicts a persistent correlation between gas metallicity and magnetic field strength. We also generate all-sky maps of the Faraday rotation measure from the simulation-predicted magnetic field, and present a direct comparison with observations.