1) There is not enough power of the rays to start the chemical process. The illumination from the full moon is simply negligible compared to the illumination from the sun. Single ultraviolet photons that accidentally hit several living cells will not be able to fix the result.
2) Redistribution of the spectrum. The moon is not a perfect mirror. The moon absorbs the entire spectrum of radiation, heats up, and, like any heated body, re-radiates energy with a maximum in the infrared range. As a result, the energy of the absolute majority of photons does not reach the red border of the photoelectric effect and cannot start this chemical process.
I wish you sunny days for tanning! Since the Moon is not suitable for us for this))