"The simplest way for you to be PCI compliant is to never see (or have access to) card data at all. Stripe makes this easy for you as we can do the heavy lifting to protect your customers’ card information." [1]
Interesting question whether Stripe incurs statutory privacy duties to the web vendor and the buyer separately. I would imagine so, because given the "triangular" nature of this kind of Stripe transaction, Stripe ends up collecting data from two parties.
[This is not legal advice]