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.
Their FAQ explicitly says that they do not currently support RFC 3156, "MIME Security with OpenPGP" (http://tools.ietf.org/html/rfc3156). I don't know this stuff well, but that makes me suspect that PGP/MIME is not currently supported.
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.
