probably OT: Is it possible to link Stripe to a credit card so I can perform an online shopping transaction on behalf of my customer without handling their credit card data?
What's your use case? You can't use cards stored in Stripe on non-Stripe sites, if that's what you're asking.
What I was imagining is stripe issuing a credit card that my system can use to perform credit card payments on any generic webshop (either manually or some web scraper) and have that linked to a payment into my stripe account by a specific customer. I am not aware of anyone offering this service.
Why on Earth would you need to be able to do that?
Sounds like an extra layer of card fraud protection might be a use case.