Many if not all of the acme clients out there store the key and certificate as a file. Just copy that to the various machines terminating the SSL connection.

In our case, there's a script on the frontend machine running HAProxy that's fetching the certificates and putting them on a shared file system so the backup machine also has access to them.

The traffic between frontend host and application servers is unencrypted in our case as it's all one rack under our control. If it wasn't, we'd be using a self-signed certificate for that connection.