Oxygen cycle is a biogeochemical cycle which involves the movement of oxygen between its reservoirs - the atmosphere, biosphere and lithosphere. ie., oxygen from these reservoirs are cycled from one to the other and back.
The largest reservoir of oxygen is the lithosphere where oxygen is trapped in the form of silicates and mineral oxides. The atmosphere contains only around 0.3% of oxygen.
The main driving factor for oxygen cycle is photosynthesis. Here comes the role of microorganisms. Many photosynthetic microbes like cyanobacteria and phytoplanktons make heavy contributions to the overall rate of photosynthesis in the biosphere. Carbon dioxide is consumed and oxygen is released during the process. Several other processes also release tiny amounts of oxygen into the atmosphere from the lithosphere and biosphere.
The oxygen from the atmosphere is used up in respiration by organisms or in decaying of matter by microorganisms. Some amount of oxygen is used us by the lithosphere too, in processes like weathering or surface reactions like rusting.
Thus oxygen is cycled between the different spheres by its production and consumption. We can see that microorganisms play a key role in both production of oxygen as well as its consumption. Thus microorganisms are an important part of the oxygen cycle.