Tried to book places Rick has suggested but not too much luck. We are looking for a decent place, don't plan on spending a lot of time in our room.
venere.com
There are plenty of convents and monasteries near the Vatican, see Hotel Alternatives in Rome. Or you can look at this post which has the top-ranked residences by venere.com and booking.com: The Best Hotels and B&B's in the Prati-Vatican Area
http://www.romabandb.it/inglese/affittacamerearoma.htm http://www.romeby.com/
Used the latter to locate a fabulous apartment. I think if you used "near the Vatican" as your criteria you'd find something to your liking. Enjoy.
We stayed at Palazzo Olivia (http://www.palazzo-olivia.it/) in an apartment, about a 20 minute walk to the Vatican. They have diffrent sized apartments and the one we were in was very comfortable. Also, very accomodating to English-speaking tourists. I would recommend it if that's what you are looking for.
We stayed at the Sisters of the Pallotine convent that is mentioned in Ron's great post. VERY close to the Vatican and the bus stop where one of the popular buslines that hits the tourist points stops. VERY CLEAN basic accomodations, twin beds, and a better than some curfew.
These folks are really good and have locations they can refer near the Vatican. http://www.cross-pollinate.com/ I think it is run by an American couple.
booking.com but read the reviews carefully.