Mozilla isn't available for Amiga (yet), so it can only be used under Linux or Windows.
As for AWeb, download the latest PL (public license) version (sorry, can't remember the site) - it's on 3.5.01 Beta IIRC, and works much better than the trial 3.4 version that comes with OS 3.9. It's not ideal, but works fine for me here (or at least it will when I've got time to sort my SSL woes, can't get into my email!).