For this to be able to send PGP/MIME emails, the webmail service would have to allow the client to uploaded a body which is not tampered with at all, and also to specify the mime type of the email. That way, you could upload a pre-built MIME structure to be used as the body, generated by the OpenPGP extension, which includes the encryted attachments.

This could work, but the web service would have to support it as well as the extension.

Then there is also the issue of how does the receiver read the message? The extension would need to be able to parse MIME, and allow access to the separate parts.