The dynamics of the ordering processes in two-dimensional lattice models with annealed vacancies and nonconserved order parameter is studied as a function of temperature and vacancy concentration by means of Monte Carlo temperature-quenching simulations. The models are Ising antiferromagnets with couplings leading to twofold-degenerate as well as fourfold-degenerate ordering. The models are quenched into a phase-separation region, which makes it possible for both types of ordering to observe the following scenario of ordering processes: (i) early-time nucleation and growth of ordered domains, (ii) intermediate-time trapping of the mobile vacancies at the domain boundaries, and (iii) late-time diffusion of vacancies along the domain-boundary network towards the surface. In the case of high dilution, the ordering processes correspond to early-time island formation and late-time coarsening and compactification via coalescence. The domain-size distribution function, which is approximately log-normal, is shown to obey dynamical scaling over a substantial time range for both types of ordering. The growth for the pure systems is found to be described by a power law with the classical growth exponent n=1/2. For the dilute systems there is a distinct crossover at late times to a much slower, possibly logarithmic growth mode. These results apply to both types of ordering, suggesting that the effects on ordering dynamics of vacancy diffusion and annealed randomness do not depend on the symmetry of the order parameter. The results of the model study are relevant for the interpretation of experiments on ordering in impure systems and off-stochiometric alloys, grain growth in radiation-damaged materials, and may also shed light on aspects of sintering processes. The finding of a crossover from an algebraic growth law for the pure system to a slower, possibly logarithmic growth behavior in the dilute system is in accordance with recent high-resolution low-energy electron-diffraction experiments on the oxygen ordering on W(112) surfaces doped with nitrogen and time-resolved x-ray-scattering studies of the ordering in thin films of Cu3Au alloys with extra Cu.