We investigate the dependence of the electronic band structure on the localized phonon mode at a Ga vacancy in GaN. The electronic states and phonon modes are both calculated using a first-principles method based on the density functional theory. Comparing the calculated electronic band structures and phonon-frequency densities of states of GaN without and with a Ga vacancy, we find that 1) there are localized electronic midgap states closely above the top of the valence band, 2) localized phonon modes appear above the acoustic and optical phonon bands, 3) one of these localized phonon modes contains asymmetric distortions in which one of the four N atoms around a Ga vacancy strongly oscillates, and 4) the localized electronic midgap states strongly couple with the localized asymmetric mode. These results indicate that a Ga vacancy can act as a nonradiative recombination center and may trigger defect reactions in GaN-based devices.