I believe the feature I'm looking for is called a "CD tray". This enables you to print an image right on the surface of a CD or DVD (provided you use a CD or DVD with a printable surface).
It seems that these printers were popular a few years ago, but are now hard to find. I have searched B&H, and am unable to find a way to filter searches for printers with this feature. I have also exhausted my local retail sources - Best Buy, Staples, Office Max, etc. Nobody locally seems to carry printers with this feature anymore.
If anyone uses this feature on a printer and is pleased with the results, please let me know the make/model of your printer.