The quick answer is that communication with Let's Encrypt all happens over an API from an automated system. So there's some software running on your servers, which is handling renewing and installing those renewed certificates wherever they need to go.
If that system isn't working, you would need to look at its logs to find the error message to figure out what exactly is the problem, to fix it and allow it to once again automatically renew and install your certificates.
There's no "authority" associated with the email address that's on an ACME account. You can change it if you want to, using the protocol. For instance, the popular client certbot has a certbot update_account command. But that would just change the location for possible future notifications. (And in general, it doesn't really have much of a difference from just creating a new account with the new email address, unless you have some rate limit exceptions or the like tied to your existing account.) And as stated above, Let's Encrypt is moving away from doing any account-specific notifications for privacy reasons, and recommends just subscribing to their newsletters if you want to stay up-to-date on what they're doing.