The kinetics of domain growth in the orientationally ordered phases of a two-dimensional anisotropic-planar rotor model with vacancies is calculated by computer simulation of thermal quenches below the phase-transition temperature. The ground state of the model is a sixfold-degenerate herringbone structure in the absence of vacancies and a twofold- or eightfold-degenerate pinwheel structure in the presence of quenched or annealed vacancies. Quenches to zero and nonzero temperature are considered. The following aspects of domain-growth kinetics are studied: (i) freezing-in phenomena at zero temperature, (ii) temperature dependence of kinetic exponents, (iii) importance of soft and hard domain walls, (iv) relevance of the number of degenerate ground states, and (v) influence of quenched and annealed vacancies. Potential physical realizations of the model include N2 and mixtures of N2 and rare gases physisorbed on graphite. The results of the model simulations are discussed in relation to recent theories of domain-growth kinetics and compared with results of other model simulations as well as experimental studies of ordering processes in adsorbed molecular monolayers.