I think you may just be confusing things by asking about if you can "issue certificates" yourself. All issuance for publicly-trusted sites comes from the publicly trusted CA. But you can certainly have one system responsible for validating names and asking the CA to issue the certificates, and then centrally distribute the certificates from there to the systems that need to use them. (Though having each system responsible for automating its own certificates can work fine too.)
Maybe you can describe more about how you're currently automating getting, distributing, and installing certificates from whatever CA you're currently using? And why are you looking at moving to Let's Encrypt? Many CAs have free domain-validation certificates, and many CAs support ACME for automation. Let's Encrypt does both, sure, and is a great place to get your certificates from, but there are other good choices available too.