I doubt they will just give you a copy of it, but it sounds like it's just storing the emails into a database, and then running a cronjob weekly to mail() it out, really simple.
It's so simple to do it that I just dunno what to start with ). It's just a basic mysql insert with (probably) two tables: subscribers and newsletters.
Here is a newsletter script if you don't what to make it your self. http://www.cgiwebtools.com/index.cgi?action=products&id=3