Okay, here's the next diagnosis.
Are you running the SAME exact versions of OS X on all your computers? Since it works on your iMac, and your 15-inch PowerBook, what system is running on your 12-incher?
Also, are there anything that you may have installed on your 12-inch that isn't on your other two computers (which may somehow be causing a conflict)?
By the same token, did you install anything on your iMac and 15-incher that your 12-incher needs in order to run the VZ Access software?
I know that's mind-boggling, but at this point, that's how to deduce what the problem is, since you have two out of three machines that work fine for you with your USB modem and the VZ Access software.
Because, 80% of the time, installing software on a fresh/new user account will work since that is a clean account without any other installations/preferences/etc. that would "corrupt" the installation process.
So that is my assumption right now. You are either missing something on your 12-inch PowerBook that is not allowing you to run it (or have something on it that clashes with your VZ access software), or your OS X version is not the same as your other two Macs.