It depends on how the website is set up. If you're taking Customer details on your own platform then yes, that data is recorded. But that data would also be sorted regardless because you're buying something.
If the platform is designed to redirect the user to a third party payment gateway I e ayden or PayPal, then no, your data is only recorded up to the point of leaving . Regardless of this all session recordig providers offer an option to grey out sensitive data. There isn't a single one that doesn't because if GDPR rules. I work in the EU and we use an American based company for this, when GDPR came out it was easier for them to implement it across their entire service than to create regional versions.