Let me first clarify a few things.
- Whether a calculation is Kohn-Sham or Hartree-Fock has nothing to do with whether the calculation is periodic or not. In fact, both Kohn-Sham and Hartree-Fock can be applied to both finite-sized clusters and periodic systems. Some programs may only support some of the 2x2=4 combinations, for example it is well possible that some program can do Kohn-Sham calculations on both periodic systems and clusters, and Hartree-Fock calculations on clusters, but not Hartree-Fock calculations on periodic systems. But then this is a limitation of that particular program, not a fundamental property of Kohn-Sham and/or Hartree-Fock. One may, however, easily have the false impression that Kohn-Sham is inherently periodic and Hartree-Fock is inherently non-periodic, if they learned the Kohn-Sham equation from a condensed matter physics course (which typically only mentions the periodic case), and the Hartree-Fock equation from a computational chemistry course (which typically only mentions the non-periodic case).
- Regardless of whether Kohn-Sham or Hartree-Fock is used, defects that are not periodically distributed can only be approximately modeled, either by using a large enough but finite-sized cluster, or a large enough (periodic) super cell. This is because you can only describe a finite number of atoms, otherwise it takes infinite amount of time just to write the input file, let alone for the calculation to finish. The cluster or super cell can however be extremely large, so large that the periodicity and finite size effects are effectively negligible, especially if you use linear-scaling DFT or machine learning methods to speed up the calculation. You can also use a series of increasingly large clusters or super cells and extrapolate the results to infinite size. (Nevertheless, if the non-periodically distributed defect is distributed in a more or less regular manner, for example following the lattice of some quasicrystal, then it is conceivable that an exact simulation is possible; however I don't know if exact simulations of quasicrystals have been reported. Even if yes, such techniques are not commonplace yet and probably involve very complicated mathematics.)
With these being said, it is apparent that both the Hartree-Fock and Kohn-Sham equations can be used to simulate defects, but only after either the crystal is truncated to a finite size, or the distribution of the defects are approximated as periodic; these two approximations can however be made very good. So which one is better for simulating defects? Theoretically the answer is Kohn-Sham, since the Hartree-Fock method lacks correlation completely, so that adding any DFT correlation term on top of HF gives us a DFT functional that is usually better than HF. In practice, however, some programs may support pure functionals and HF, but no DFT functionals with 100% HF exchange; in this case, it may turn out that all functionals supported by these particular programs perform worse than HF, especially for those defects where self-interaction error plays a significant role. However, again this is a problem of the program, not a problem of the theories.