It depends on whether they use PGP/MIME or inline PGP. Without installing the tool, I'm guessing they use PGP/MIME, because you pretty much can't use html email with inline PGP, and I doubt they're going to default to plain/text mails.
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.