Not sure of the exact requirements, but to get access to Lincoln Center you have to be a working theater professional and working on a project that would pertain to whatever show video you wish to view.
As was posted to Roxy, us great unwashed masses are not really welcome to just peruse the video library (oh, that we could).
Try looking on YouTube which has full shows clandestinely video'd on a phone, old relics from prior television broadcasts (Applause was the first one I had found) and every Tony Awards show in its entirety from first TV broadcast to the 2014 show).
One other video suggestion, there is also the Museum of Television and Radio on 52nd Street. It has a wealth of videos of all sorts, and some pretty good Broadway or theater related shows. That is a great way to spend a raining afternoon when visiting NYC.