The Vatican sells an interesting assortment of coins and coin sets honoring Popes, Papal topics, and other subjects of interest to Catholics. However, the online Vatican coin store is very hard to find unless you know Italian, and they do not take orders over the Web. In order to buy these coins, you will need to call, fax, or email the Vatican's numismatic office to see what is currently available for sale. The Vatican does have a listing of a few of their Euro coin sets online in English (along with the prices in Euros) but to access the full inventory, it is best to email them (and yes, someone there will reply in English.) Some of the coin sets shown on the Vatican Web site have been sold out for years.
How to Reach the Vatican Coins Department
This is the direct link to the Vatican Coins and Stamps Store entry page. Click on the link that says Numismatics to see the few coin sets listed. If you hunt around a little, you can find some miscellaneous coins and medals listed, but most of them have been long sold out.
You can contact the Vatican's coin sales office at firstname.lastname@example.org, or call them at 011-39-06-6988-3708. Keep in mind that their normal business hours will be about five hours ahead of those in the U.S. Eastern time zone.
The Vatican State Web Site
The Vatican State government also maintains a Web site, which has a coin-selling store on its site, but the store has never actually opened for business. It lists many coins, medals, and sets which are different from the ones on the Vatican Web site. If you are eager to buy these coins, perhaps check back every few weeks to see if this store ever opens (and if it does, please let me know!)
Special thanks to Bill from New York for submitting this question and providing information about the Vatican State government Web site.