I've never tried to do anything like this, but from the user end of Quark, I imagine automating the process would be kind of hard. The reason is that Quark doesn't really distinguish between elements of the same basic type. I don't see how an automated solution would be able to tell the difference between stories, headlines, breakout boxes, text shadows, flags and all the other textual elements of a newspaper, many of which wouldn't be treated alike by a Web site and some of which probably wouldn't appear at all.
I've spoken to people who actually have companies dedicated to this (like College Publisher, which provides Web sites for school newspapers), and they use the online form too, but the stories go into a database on the server rather than a static HTML file. So probably your best bet is just to modify that interface that they're used to.