I plan to write a web application using the Sinatra framework, and want to add HTTPS to it using Let's Encrypt.
Sinatra is merely the framework you use to build a web app on top of a web server. The only thing that matters when using a Let's Encrypt client is that you're using a support web server such as Apache or Nginx.
The documentation lists out these requirements: https://certbot.eff.org/docs/intro.html#system-requirements
You can use
certbot regardless of the web framework you choose as long as you're running on a supported web server.