Do I inform my Professor that our digital exam isn't recording us?
From your comment, I note your concern that if you report the matter, you might be required to sit the exam again.  I think that is unlikely.  It is unlikely that your professor would impose additional work on the students due to a technical error that was not their fault.  It is more likely that your professor would be glad to have been informed of the technical problems, and would use this information to try to have the technology run correctly in future exams.  It's really up to you, but as an academic with experience in teaching, it is certainly useful if students report problems, so that you can improve future iterations of your teaching.