Do you need to reinstall or only need the serial number?
To retrieve the serial number - if Aperture is still running on any Mac you own, you can see the serial number in the "About" panel when Aperture is running:
From the main menu bar: Aperture > About Aperture.
If you do no longer have a running Aperture installation to retrieve the serial number, contact Apple with a proof of purchase and ask for a replacement serial number, see these links as to what is considered to be a proof of purchase:
Pro Application Replacement Serial Numbers: http://support.apple.com/kb/HT1861
If you need an installer to reinstall Aperture without CD/DVD, download the Aperture Trial:
You can download the Aperture Trial here:
Delete your current Aperture version from the Applications folder, then run the Trial installer and immediately update using "Software Update" from the -menu.
Then launch Aperture and enter your serial numbers, all while signed in as an administrator.