Upvote:3
I think this is one where you have to contact the hotel and ask whether they are willing to charge your card through the booking site, (which they can in case you do not show up.)
Different hotels have different ways of handling this kind of request and while some will make exceptions for 'emergency type' circumstances, others will only work by the rules set by the management.
Most hotels I have stayed recently insisted on getting a card to record (again) which does not need to be the same card as the card you made the reservation on. But they can charge your current card (if it does not get blocked by the bank for being missing.)
Many hotels even accept (or prefer) cash payments. So if you can bring cash that might be a way out.
(As I posted in a comment to the question, check with the bank/credit card company that it is late and not missing. And be prepared for the card getting blocked if someone else starts using it or if they assume you are not the one using it. It is risky to activate a card you do not hold in your hand, just for the reason of it possibly being stolen.)