Yes, Moon once and continues to support life.

The giant-impact hypothesis, sometimes called the Big Splash, or the Theia Impact suggests that the Moon formed out of the debris left over from a collision between Earth and an astronomical body the size of Mars, approximately 4.5 billion years ago, in the Hadean eon; about 20 to 100 million years after the solar system coalesced. ~ wiki

The earliest evidences of life on Earth are biogenic carbon signatures and stromatolite fossils discovered in 3.7 billion-year-old metasedimentary rocks discovered in western Greenland. ~ wiki

When the Moon formed there wasn't known life on Earth so life wasn't transferred to the moon in its formation.

The Moon, as a stable surface celestial object couldn't and can't support unprotected life. Mainly because it has such a minute atmosphere (yes it has an atmosphere), too little gravity, no protection from cosmic or solar radiation and no free water (due to temperature).

The common elements of life that are throughout the solar system may be present but life can't form because the complex chains needed for DNA and RNA, among many other things, can't be sustained long enough.

Life can exist on the Moon in constructed habitats. I'd be curious if water bears could be exposed to the Moon and then retrieved and revived?
I would also be curious if certain seeds could survive exposure and be brought back to a habitat and grown?

Highly unlikely.