You can do that by first getting a gift card or reloadable cash card from a store that sells phones - for example Best Buy or Target. Then you can order phone service on-line by using the card.
Letstalk. com has free cell phones, also check with the different cell companies, they always have deals based on your contract term on phones. They do credit checks for service though.
Anyone can now purchase a cell phone without a credit card. There are Boost phones and Trac phones available in K mart stores. This is sometimes a better purchase for young teenagers, so no monthly bill is required.