If you use the “Login with Google” option to log into an account, these logins will stop working. You will need to transition those third-party logins to a personal account/email address that you have access to.
If you login using a username/password combination where the username is the email address, this will likely continue to function. However, when you no longer have access to the Vanderbilt email used for the account, you will stop receiving those emails and that could prevent login as sometimes the email address is used for things, such as multi-factor authentication and password resets.
If you have issues with transitioning these account logins, you should contact the third party for support.