I use Amazon quite a bit without a credit card. My husband is adamant about 'no credit card's over the internet'. I see his point, but really, anyone who knows what they're doing can hack into our personal info, if they really want to. So much stuff is on computers, and it's supposed to be protected, but there are those who can find their way into anything! Eh--- whatever makes him happy!
But anyway -- you can send them a check, or money order. It takes a little bit longer, especially if you send a check, because they have to make sure it clears. But every time I've done it, my stuff has arrived a lot quicker than what was estimated.