Sorry if this is a dumb question, but why? If I'm not mistaken, Let's Encrypt supports validation via DNS now so you don't even need to have a working webserver to issue a certificate. Automating a script to perform a renewal should be much simpler than headless Chrome!
If your DNS provider doesn't have an API, that seems like a separate issue but one that is well worth your organization's time if you're working in the enterprise!
I'm really not a fan of it but I'm happier paying for a one year cert than doing that