Wow. Curioser and curioser.
Hmm. It's possible that there is some kind of proxy setting in the Macs or browsers that won't work. It's also possible that the CGI for the system looks at the browser's referrer and returns nothing for browsers it doesn't recognize--which would be a very weird thing to do.
Try this, just for kicks:
On one of the Macs where Safari doesn't work, go to Safari's preferences, click Advanced, and then click the Show Develop menu in menu bar (assuming you don't already have it turned on).
Then click Develop->User Agent->Firefox 3.5.3
Then try to go to the Web page. If it works, then what's happening is that the script is written to return a phony 404 error if it does not recognize the browser.
As a side note, these Macs that do and don't work are on the same network, right?
