You can compress using a couple of things you already have...
right-click and choose Archive should zip a file or folder, I think even on OS X 10.3.
Or if you are sharing with another Mac user, you can use Disk Utility to create a compressed DMG.
If you absolutely need to create ZIP files to send, then I would suggest Stuffit or Stuffit Deluxe. Email them and ask if they have an older version that will work on your OS:
www.stuffit.com
(the current version won't, but you may be able to buy the current version and get access to older installers)
Just saw this:
http://www.stuffit.com/support/updates.html
10 10.0.2 English StuffIt Standard 10.0.2 Mac 9.3 MB .dmg This installer will require a StuffIt Standard 10 serial number in order to not expire, but will act as a 30-day trial if not registered.
For Mac OS 10.3-10.3.9
Not sure if that will include zip format, but you can certainly check it out.