That varies game by game, some will let you without a video depending on the time and previous records, eg Mario Galaxy, however smaller games with only a few runs may require a video, especially runs in the top 4 places. If you're unsure if you'd need a video or not, I suggest heading to the specific games' forums and asking there.

In all honesty you're better off not even considering submitting without a video and just recording with a webcam resembling a potato if that's what you gotta do proofless runs only ever lead to problems

For the games I moderate, I require video for everything. It's not that hard to record, even if you just aim a camera at the screen.

As mentioned before, it's entirely on a game-by-game basis. As a general rule though, WR stats should NEVER be without video.

Do keep in mind that you have to tell twitch to save your VOD's. So if someone is just starting PS4 or Xbone console streams, that's an important thing to remember to point out.