You can only be 'signed in' to one Altium Vault at any given time, since you are connecting to a platform of services, rather than to a single Vault service. Typically an organization will have a single Altium Vault installation. However, if more than one Altium Vault is available, simply sign out of one, and sign in to another, as required.
You can of course still connect to older (legacy) Vault technologies - Altium Vault Server (installed on another PC), Altium Personal Vault, Altium Satellite Vault - through the Data Management - Vaults page of the Preferences dialog, in Altium Designer. From the Sign in button's associated drop-down menu, choose the Add Vault (legacy) command.
While legacy Vault technologies can still be connected to, and used, there is no further development to these types of Vault. You will therefore not enjoy the current enhancements found in, and future improvements gained by, using the latest Altium Vault.