Game seems to install. But it stops, displaying the error message: "Your game requires a system restart to play. Please restart your computer. If this issue continues, reach out to our player support team.". Actually, this is caused by the anti-cheat software that is needed to run the game, called Vanguard.
It turns out, Vanguard has a kernel driver component designed to run in Ring 0 of the Windows kernel, the highest possible access level Windows offers. Software that runs in Ring 0 has unfettered, admin-level access that allows it to scan files and make changes without the user’s knowledge.
So it is unlikely that Valorant will every run under Crossover.