[schoen]

Let's Encrypt implements a protocol, or, if you prefer, an API called ACME, which is being standardized at IETF. This allows you to make software-based requests for certificates. You can implement an ACME client or adapt an existing one to make requests for certificates on behalf of your customers.

See

https://letsencrypt.org/docs/integration-guide/

There are a lot of clients, several dozen of them already. There isn't necessarily a client that's specifically oriented toward provider integration, but one of the lightweight clients like acme.sh might be a good fit because it will work well with external scripting.