It depends on the type of API provided by the payment gateway processor. Some need the Details and actual processing to be done at there end and some does allow to process the payment despite not logging in.
As for Paypal goes they have APIs which allow you to process payment without visiting their website and GoDaddy does exactly that.
But there are lots of Issues with the kind of implementation you are looking for which I think I should share with you.
You would be storing the CC Details on your server / database which means that you need to have the server hosted at your place. You cannot have a dedicated server hosted in some hosting provider and do this as that could lead to problems when it comes to fraud protection.
You are trying to guarantee your customer what could be a security issue when it comes to security of your hosting provider.