No, you don’t need video and screen recordings.
Long Answer
There are no hard requirements for user testing. At minimum, you need some sort of proof that you did it, which steps were performed and any usability problems you encountered (and whether you solved them). The minimum viable implementation of this would be a simple protocol per user test which includes the steps performed, whether they were performed successfully and which problems were encountered.
Video and/or screen recordings are nice to have for your internal purposes but not a must from a regulatory perspective.
Don’t forget to have a sufficiently large number of users (at least five).
Good luck! :)