You should be able to do this with most payment gateways that support recurring payments.
Typically what you would do for new customers is capture their card details 'as normal' (via the payment gateway). The payment gateway will return a token id which you then store against the customer record
Next time that customer makes a payment you can submit the tokenised card number to the payment gateway
Since you mentioned Amazon, you might want to look at their payment service API: http://aws.amazon.com/fps/