The Internet Archive (IA) routinely takes snapshots of websites, indeed of much of the content on the web. Their service can be used to demonstrate the content of the site on the date of any such snapshot. They also have a procedure by which anyone can request a snapshot of any web page at any time. Although they do not guarantee to comply with such requests, they often do. It has become common practice to use the IA to establish, in court cases, the content of a web site on a particular date.
Their FAQ page says:
How can I get pages authenticated from the Wayback Machine? How can use the pages in court?
The Wayback Machine tool was not designed for legal use. We do have a legal request policy found at our legal page. Please read through the entire policy before contacting us with your questions. We do have a standard affidavit as well as a FAQ section for lawyers. We would prefer that before you contact us for such services, you see if the other side will stipulate instead. We do not have an in-house legal staff, so this service takes away from our normal duties. Once you have read through our policy, if you still have questions, please contact us for more information.
I recall reading the IA FAQ for lawyers in the past, but am not able to find it just now.
Note that even if you establish what their policy was when you signed up, you will still need to determine what actual rights this gives you. Most privacy policies are carefully written so that they will not give users more rights their the site owners choose. And n=many such policies contain language such as "the site may change this policy at any time without notice. Continued use of the site constitutes consent to the revised policy." Of course, if the EU GDPR applies, such a disclaimer may not be valid.