The dynamics of the 2D Coulomb glass model is investigated by kinetic Monte Carlo simulation. An exponential divergence of the relaxation time signals a zero-temperature freezing transition. At low temperatures, the dynamics of the system is glassy. The local charge correlations and the response to perturbations of the local potential show aging. The dynamics of formation of the Coulomb gap is slow and the density of states at the Fermi level decays in time as a power law. The relevance of these findings for recent transport experiments in Anderson-insulating films is pointed out.