If someone were to die on the moon, would their body decay?
I am assuming in spacesuit here, on the face (lit side) of the moon. Bodily degradation involves much more than external fungi and bacteria.

Cells that receive no oxygen or nutrients die. We talk of such tissue death as dry gangrene when it affects extremities, such as fingers, feet, etc. However, we also recognize gangrenous bowel, etc. which results in tissue necrosis. 

Such necrotic cell death is the consequence of acute disruption of cellular metabolism, leading to ATP depletion, ion dysregulation, mitochondrial and cellular swelling, activation of degradative enzymes, plasma membrane failure and cell lysis [1]

Lysis is messy and wet. Combined with the fluids in our bodies, what one would end up with is a mushy, smelly degraded body, not a preserved body. For a while, anaerobic bowel bacteria would flourish (which smell terrible). 

Add to this the extremes of temperature (253° F in the sun and -243° F in the dark.) The suit would have lost it's heating and cooling mechanisms, so the body would alternately spend 14 days in the heat and 14 days in the freezing cold depending exactly where it was (lets say the equator of the moon.) These freeze/bake cycles would further contribute to degradation through ice crystal formation and thawing.

Eventually, because there was no new substrate, degradation would come to a halt, but I'm not sure at what stage this would be. I assume, though, there would be a vast difference between a mummified body (done by dehydration) and a body left to degrade in a spacesuit.

[1] The mitochondrial permeability transition in cell death: a common mechanism in necrosis, apoptosis and autophagy, John J. Lemasters et. al, Biochimica et Biophysica Acta 1366 (1998) 177-196