No, CoD: Warzone does not run through Steam. In fact, the new CoD: Modern Warfare isn't even available on Steam if you wanted it. Warzone is downloaded and launched through Blizzard's Battle. No you need a Blizzard Entertainment.
Installation on PC
1
Launch the Battle app and log in to your account.
2
Select All Games.
3
Find and select Call of Duty: Modern Warfare II.
4
Select the Install button to download and install the game.