Pokémon Legends: Arceus is a mainline entry in the Pokémon series, and while it retains many of the same features as the other mainline games, it also differs in some ways. While the most noticeable difference is the gameplay, the way the story works and the player's goals are also different. This includes gym battles.
If you were wondering if there were gym battles in Pokémon Legends: Arceus, the answer is no. The Hisui region is before the Battles of Sinnoh and Gym. However, there are directors who act as gym leaders. Players will challenge these guardians, not for gym badges, but rather for the right to challenge Noble Pokémon.
Related: What are Guardians and Noble Pokémon in Pokémon Legends: Arceus?
This isn't the first time The Pokémon Company and Game Freak have removed Gym Battles. Pokémon Sun and Moon removed this feature and instead used Kahunas Island. Taking this route adds a breath of fresh air to the games and makes them more unique.
Besides the fact that players will face Guardians, it's unclear exactly how these challenges will play out. Gym battles featured other trainers and puzzles before the player could challenge the gym leader. Only time will tell how this is handled in Pokémon Legends: Arceus.
For more information on Pokémon, see Jubilife Village from Pokémon Legends Arceus related to Jubilife City from Diamond and Pearl? and How to Get Darkrai and Shaymin in Pokémon Legends: Arceus on Pro Game Guides.