VaultWarden is a reimplementation of BitWarden’s server. The clients are compatible.
Anyway, you can add your TOTP codes to a Bitwarden item. Either scan the code or enter the key manually.
It’s up to you if you want to risk storing the TOTP alongside the password, as it means you lose the second factor if your vault is compromised. For a random site? Sure why not. For financial/important stuff? Maybe not.