I have a checkout process for a shopping cart that is currently storing credit card data in the session for retrieval once the user finalizes the purchase. The purchase process
What country is this based in and what credit card companies are involved? The entire approach of actually sending full credit card numbers back to the client (in whatever form) makes this sound like you have not dealt with professional credit card handling (please don't take that as an insult).
Is your client willing to run afoul of Visa/MasterCard/AMC/Discover's collective rules for online credit card processing (PCI DSS)? Your client could end being barred by the major credit card companies from doing transactions with them. In general it is a very bad idea to try rolling your own online credit card handling solution - it's worse then rolling your own cryptographic algorithm, as there can be serious fines applied to your client (fine print in their merchant agreement). A true PCI DSS solution requires tens of thousands of dollars in certifications and audits to ensure it handles credit card data in a truly secure fashion - this is why almost everyone uses an existing online processor.
What about using TempData? You'd need to put the value back into TempData between the confirmation and finalization actions, but at least it will be discarded with each request. Note that TempData uses the Session for storage so it's no more secure while it's being stored, but it does have the automatic removal feature. I, too, would resist storing the number on the page. I suspect that this violates the PCI rules.
Another alternative would be to store the card info in a database. If you keep it at all in your application you're probably already subject to the PCI rules anyway. Storing it in the DB makes it easier as then you only need to put the billing card id in subsequent requests. This could easily be in a hidden field.
The best way to handle this scenario is to use a payment service that supports two things:
Authorization allows you to reserve the designated charge amount at the time the payment information is received, and then Completion allows you to commit the charge to the payment batch once the payment/order is confirmed. If the order is canceled, you don't have to issue a Completion and you can also attempt to delete the authorization as well.
Regarding tokenization, most gateways that support the aforementioned method of handling payments will return a token, typically a numeric id, for the pending transaction. The token may then be handled however you wish, as it has no value to anyone without access to your authentication credentials at the gateway. This transfers the burden of safety to the gateway as well.
Storing the actual credit card information in any way other than relaying a charge to a gateway/processor is a bad idea. Beyond the problems of securing the data, this will also put your application into card information storage scope for PCI/PABP, which entails a lot of rules and regulations that you won't want to deal with in your application. I believe there is also a regulatory fee that will be imposed in the coming year for compliant applications, reputedly $10k USD. All of this is likely not worth the trouble to you or your client.
Last, during intermediate processing (in-page/event/route handlers), you can use a SecureString to hold the contents of the data until you no longer need them.
SecureString class (System.Security) @ MSDN