Your request is more of a Zimbra “How To” question than a LetsEncrypt “How To” question/problem.
You will first need to find out how to install public certs for multiple domains in your specific Zimbra version.
Zimbra runs on several flavors of Linux and uses OpenSSL (these are two good things which play well with LetsEncrypt). But Zimbra is also very specific about where things go and how they are manipulated; and things have changed much as the versions have changed.
Once you get the hard stuff answered, update this post if you have any LE implementation issues or findings to share for others that may travel that same path one day.
A) Obtain certificates this forum can help with approaches etc (if you articulate your setup and how you plan to do this). If you don’t know then a general google of let’s encrypt + zimbra should give you a good starting point. Alternatively you can search zimbra on this forum as it has been discussed before