They found a fungus in the remains of the reactor at Chernobyl that had adapted to using melatonin to convert ionizing radiation to to produce sugars so in principle a life form could evolve that would view the high radiation environment of the moon as smorgasbord.
Despite all the talk about anti-oxidants, free-radicals make biology go around, they just have to be controlled. If ionizing radiation was being actively "eaten" for lack of another term, then likely it wouldn't damage the rest of the organism to badly.
There are extremophile organism from the archaea domain that survive massive extremes of pressure and heat. Some of them that live deep in minerals veins use virtually no water.
So, it might be conceivable for a microbe to evolve that ate ionizing radiation and required little water. Likely, it would live just under the soil and hibernate in the lunar night when everything froze and the radiation influx died down.
Boring but plausible.
Then they spontaneously organize like slime-molds and searching for water, which the lunar base just happens to have a lot of.
Radiation eating blob immune to extremes of temperature and you have to get every last cell to kill it. I'd watch that movie.