In a blog post on the official Call of Duty website, Sledgehammer Games announced that players would be able to do this in COD Vanguard's Multiplayer and Zombie modes, as well as Warzone (but only after Season 1). Here's how the new Operator XP system will work in COD Vanguard.
How does the COD Vanguard new Operator XP system work?
According to the blog post, "all 12 Operators at launch and others to come — can be levelled up to unlock new Skins, including special Gold attire, Weapon XP for specific weapons, and more." In addition to this, the developers note that players can "use their Favorite Weapon to earn bonus Weapon and Operator XP."

This means that COD Vanguard players will be able to level up each of their Operators simply by playing with them, unlocking new exclusive cosmetics at given levels. This will apply to the 12 Operators available at launch and future unreleased Operators.
The developers have not indicated what the maximum Operator level will be. Still, we speculate that there will likely be some tier process in place once the game officially goes live.
It would also seem that the progression players make with each Operator will depend on various factors. For example, using an Operator's "favourite weapon" could yield bonus XP toward their Operator XP progression. Similarly, players may also be awarded bonus XP toward their Weapon XP progression as well.

On that note, COD Vanguard will ship with Weapon XP attachment unlocks and Camo Challenges, noting that players can "use attachments across all modes and unlock two sets of striking cosmetic camouflage rewards" either through Zombies, Multiplayer or Warzone.
Sledgehammer Games also revealed that the Seasonal Prestige system would return. There will be 55 Enlisted Ranks before players can unlock all Loadout items.

"Essentially, every time you play Multiplayer, Zombies, or Warzone Pacific, you'll unlock something new or at least progress toward it. And that doesn't even include the rewards available during the seasonal Battle Pass system, another way to earn content at the start of Season One," Sledgehammer Games concluded.
Call of Duty: Vanguard is set to release on 5th November 2021.
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Featured image courtesy of Sledgehammer Games.