I had to implement a corporate version of this; they had a backing account and credit line, so this was an integration that involved four parties: the vendor, the owner of the credit account, the bank backing the credit account, and the card distributor. It was a messy process, involving key exchange and SOAP requests - very much a back-channel API more than a public-facing one.

I think a more publicly-friendly API might be created if it's merely the same credit card company providing temporary virtual credit cards that are backed by your main credit line. VCCs like that are already in use, the question is merely one of how freely accessible they are through programmatic means.