Can we integrate CiviCRM with PayPal so that any purchases through our website are recorded on the CiviCRM donor info?
Welcome to SE. You can integrate PayPal with CiviCRM - See Payment Processors in the documentation (https://docs.civicrm.org/user/en/latest/contributions/payment-processors/). This will mean you can take payment for any CiviCRM contributions via PayPal and it will be recorded against the contact. This includes memberships, events and donations and you can link this from a web page. However, if you have existing purchases via some other package on your website using PayPal, it won't show these. Just those originating in CiviCRM.