The Moon Stone in Pokémon Legends: Arceus serves just one purpose in the game: evolving Clefairy.
This means that you’ll only need one stone in order to complete your Pokédex in Legends: Arceus, since it only evolves Clefairy into Clefable. You can use the stone at any point on Clefairy; there is no level requirement to be able to evolve it. This is good for completing the entry into the Pokédex, although you might need some time with both Pokémon before your research level reaches 10 with them.
Among other evolution items, the Moon Stone can be purchased from the Trading Post in Jubilife Village. It can be purchased for 1000 Merit Points, which can be gained by returning lost satchels to their rightful owners. Each one will award up to 100 MP per satchel, meaning you’ll likely need more than 10 satchels to be able to afford one Moon Stone.
A few other ways you can find Moon Stones include using Ursaluna to dig them up in the wild or finding them in space-time distortions. The Clefairy’s Moonlit Dance request also grants one Moon Stone after completing it.
If you’re not keen on running around in space-time distortions, the best way to get the Moon Stone is by grinding satchels for the MP. The request that rewards the Moon Stone isn’t available until later in the game, so early on, you’ll either want to grind satchels or run through space-time distortions to evolve your Clefairy.