Would cert-manager for Kubernetes be approximately what you’re looking for? In that particular case, it’s a service that runs in a K8s cluster, that can auto-request CSRs and send them to Lets Encrypt using one of the challenge types. It takes care of managing the returned certificates, such as renewing them before expiration and registering the private keys within the K8s key store.
I presume that you’re not already using K8s, so maybe that project will help you find the keywords to describe the sort of certificate manager that you want.