This solution is based on the letsencrypt-webapp-renewer. It uses the same core library than the Azure Lets Encrypt site extension, but it is run as a WebJob. It can (should) be installed on its own web app, and supports multiple target websites.

The author of the letsencrypt-webapp-renewer has made thorough instructions, so I won't copy them here. When granting the service principal rights, you may want to only add Website Contributor and Web Plan Contributor instead of Contributor rights. The only thing needed in addition to those instructions is the support by the web app itself.