I remember back in Pokémon Black that when you beat the game and had a certain amount of Pokémon in your Pokédex, Professor Juniper would give the Shiny Charm. Is there an item like that in Pokémon Sun and Moon?
Yep! There's a Shiny Charm in Pokémon Sun and Moon. This article describes how to obtain it:
It is important that you need to complete the Alolan Pokedex before you can acquire the Shiny Charm. The move would have you capture 300 Pokemon and register them before you speak to the director in HeaHea City on Akala Island to get your Shiny Charm.
This article goes more in depth about where exactly he is located:
He's located in a building to the left of the Dimensional Research Lab up on the second floor and wearing a gray shirt with orange shorts. If this is the first time you're speaking to him, you'll have to initiate conversations with him several times in a row until he stamps every page of your passport. Eventually, he'll give you one final stamp for completing the entire Alola Pokédex. As a reward, he'll also give you the Shiny Charm.