when matter i'm plants and animals decay, microorganisms responsible for the decomposition process are doing cellular respiration

Yes, when the plants and animals die, decomposers such as bacteria and fungi etc which are microorganisms feed on these dead bodies of plants and animals and broken down their bodies into simpler substances. This decomposition occurs both with oxygen and without oxygen. When energy is released with the addition of oxygen, this process is called aerobic respiration while when energy is released without the use of oxygen, this process is called anaerobic respiration.