The exact procedure probably depends a lot on the webmail service provider and/or webmail software... Most of them require you to go through several hoops (i.e. pages, dialog boxes, etc.) to download the attachment. WWW::Mechanize would probably be a good start to automatize the steps.

But before going into that, have you made sure that you really can't access the mail in any other way, e.g. POP3 or IMAP?