The Lord of the Rings: Gollum likely won't include Frodo Baggins, based on the timeline that it's using for its story. There have been many video games based on The Lord of the Rings and The Hobbit released over the years, many of which have included new characters, or let the player make their own character, which helps sidestep the issue of the hobbits themselves not being suitable for the killing machine role that many video game heroes have.
Gollum has an even harder task ahead of it, as Gollum is the protagonist. Gollum is a narrative-based action-adventure game, where the player must contend with the dueling aspects of Gollum's personality, as well as all of the beings that want to do him harm. Luckily, Gollum is a resourceful and sly being, who managed to escape from some of the most secure places in Middle-earth. The Gollum video game will be more about exploring Gollum's character than slaying orcs and goblins.
It has been confirmed that some existing characters from The Lord of the Rings will appear in Gollum, including Gandalf and Thranduil. It might seem logical for Frodo to appear, as he was the protagonist of The Lord of the Rings. However, it's highly unlikely that he will show up in Gollum.
LOTR's Gollum Doesn't Meet Frodo Until Long After The Events Of The Game
The developers of Gollum have confirmed that the events of the game take place before the events of The Lord of the Rings books. This checks out with the information revealed so far, as he was interrogated by Gandalf and later given over to the elves of Mirkwood, who are ruled by Thranduil. Frodo saw Gollum for the first time from a distance when the Fellowship was being tracked through the Mines of Moria. The two properly met for the first time during the events of The Two Towers. These events happen within the timeline of The Lord of the Rings. It wouldn't make sense for Gollum to see Frodo before this point, as Gollum never made it past Moria, so he never got close enough to the Shire to encounter Frodo.
It's possible that Frodo might appear in a post-credits scene, as a way of setting up a sequel that takes place during the events of The Lord of the Rings, but it's unlikely that he will have a role in the main story of the game. It's more likely that there will be a flashback scene involving Bilbo, considering how important the Riddles in the Dark scene is to Gollum's backstory, but it's doubtful that Frodo will show up.
The Lord of the Rings: Gollum will be released for Nintendo Switch, PC, PS4, PS5, Xbox One, and Xbox Series X/S in 2022.