How can I record sales data from Stripe Checkout and then restrict access to content based on the item purchased?
My understanding of Stripe Checkout is that it doesn't actually charge the user's card. It simply authorises it, and returns a token which you can then use in a server-side script to capture the payment using the Stripe API.

As such, you'll need a server-side component, in the form of a plugin. Charge might provide a decent starting point for that.

I don't think Charge works with Stripe Checkout as standard, but it's probably worth having a chat with the developer (Joel Bradbury) about the costs of adding this functionality.