They certainly don't use MySQL nearly as much as a chat script. There's no mention of it in the new TOS either. I don't have a good record of being right about these sorts of things, but I'm certain that you may use a shoutbox on your site
I'll update this since I remember the behind-the-scenes discussion on it.
Shoutboxes are now okay, with one minor exception: you are still at risk of the High Resource Usage suspension if it's too resource intensive.
That said it's no longer an instant-permanent-suspension, so it's definitely a step up since most of the good ones won't cause a HRU anyways