That's actually a potentially good business idea - a legally certifiable archiving software that captures the content at a URL and signs it digitally at the moment of capture. Such a service may become a business requirement as Internet archivability continues to decline.
I don't know how exactly it achieves being "legally certifiable", at least to the point that courts are trusting it. Signing and timestamping with independent transparency logs would be reasonable.
Any vendor who you work with should make it trivial to access these docs, even little baby startups usually make it quite accessible - although often under NDA or contract, but once that's over with you just download a zip and everything is there.
That's what I thought the first time I was involved in a SOC2 audit. But a lot of the "evidence" I sent was just screenshots. Granted, the stuff I did wasn't legal documents, it was things like the output of commands, pages from cloud consoles, etc.
What I would not do is take a screenshot of a vendor website and say "look, they have a SOC2". At every company, even tiny little startup land, vendors go through a vendor assessment that involves collecting the documents from them. Most vendors don't even publicly share docs like that on a site so there'd be nothing to screenshot / link to.
That is: if it's not accessible by a human who was blocked?
Having your cake and eating it too should never be valid law.