Can Sawmill be configured to automatically analyze the access log for my site on a shared server once a day at a given time?
Yes, if you run it stand-alone, or if your server has a scheduling program.
It depends on your web server. If you run Sawmill as a stand-alone program (rather than as a CGI program) on your server, then you can use Sawmill's built-in Scheduler to do this. If you can't run it stand-alone or don't want to, then you can still set up automatic database builds if your server has its own scheduling program (like cron or Windows Scheduler).