The Shiny Charm is not new to Pokémon Legends: Arceus, and functions just like any other charm. The only difference is just how difficult this one is to get, and will be one of, if not the, last thing you get in the entire game. Only go after this item if you're ready to really commit to becoming the best trainer possible.
Step 1: Complete the Pokédex. That's right, you will need to complete the Pokédex to get the Shiny Charm. What does completing the Pokédex mean? First, you need to not only catch all 242 Pokémon registered in the Pokédex, but you also need to reach research level 10 on every Pokémon. That means completing enough research tasks for all 242 Pokémon to mark it as Complete, indicated by a Pokéball icon.
Step 2: After that monumental task is complete, take a trip back to Jubilife Village and go into the Galaxy Hall to speak with Professor Laventon. A short scene will play out and you will be rewarded with the Shiny Charm.
Step 3: Equip the Shiny Charm and get Shiny hunting!
Normally, your odds of seeing a Shiny are 1 in 4,096. That drops to 1 in 2,048 for every Pokémon you reach research level 10 for, which you will need to do for all of them to get this charm, or further down to 1 in 1,024 if you do every research task for a Pokémon. If you stack the Shiny Charm on top of all this, your rates for finding a shiny is either 1 in 819 for a Pokémon at level 10 research, or 1 in 585 for perfect research levels. These can go just a bit lower, too, if you encounter a Mass Outbreak.