I realize radiation and extreme temperatures would probably destroy life on the surface of most planets and moons, but could life exist beneath the surface (e.g. like earth worms on earth)?
That's one of the big questions. ESA scientists, at least, think it's worth looking for underground life. See ExoMars mission.
More likely than earth worms are microbes, since some microbes on Earth live under similar conditions as presumed for underground Mars.
I asked a question on the same topic few months ago : Could any known, living organisms on Earth survive on Mars?
The answer from
dotancohen pointed the Tardigrade that can live in extreme conditions.
The other life forms I mentioned may also be resistant to extreme conditions.
Some of our gas giant's icey moons are thought to have liquid water interiors. Tidal flexing is thought to be a source of heat.
While such locations might be habitable to some (of Earth's) lifeforms, for life to actually exist there it must also be an originable location for life (or somehow had life transported to it). The origin of life on Earth is a big unknown. For sure, Earth was a very different place when it happened and it could've happened underground.
And define "life". It's conceivable that lifeforms exist that are for example silicon based, don't need oxygen, and can survive in hard vacuum, even directly use gamma and X-rays the way plants on earth photosynthesise visible light.
Such could well survive there, even thrive. But would we even recognise them as alive?