There is no “official” web GUI for LetsEncrypt; it’s designed to be run automatically with some client software on your server (see https://letsencrypt.org/about/). There are, however, lots of clients available; you can find a partial list at List of Client Implementations. You may find that one of them is compatible with your requirements.
When using a LetsEncrypt client (including gethttpsforfree), LetsEncrypt asks for an email address. I believe it’s optional, but it is requested. I understand they’ll send notifications (like of a certificate about to expire) to this email. You can enter any email you want, but it should be one that works and that you control. It need not match anything else.
You really don’t need to trust https://gethttpsforfree.com/, as you aren’t giving them anything sensitive–only an account public key and a CSR (which includes your site public key). Neither can that information be used to obtain or derive anything sensitive (like a private key).
Keep in mind that, at least as of right now, certificates are valid for 90 days, so you’ll need to get a new certificate at least that frequently.