Layer-by-layer sputtering of Si surfaces by low-energy Ar ions is studied by using scanning reflection electron microscopy. Two types of layer-by-layer etching (reversal of step-flow growth and two-dimensional vacancy-island nucleation) are observed. These results imply that vacancies created by ion impact can diffuse on the surface, and are annihilated at step edges. The diffusion length of vacancies are estimated from the denuded zone width of vacancy islands formed along steps by thermal heating after the vacancy introduction. The obtained activation energies of vacancy diffusion are 3.0 +/- 0.2 and 2.3 +/- 0.2 eV for Si(111) and (001) surfaces, respectively. We also propose an application of vacancy islands in nanofabrication, where vacancy islands are used as a self-organized template to provide nucleation sites of quantum dots.