I was thinking about running Bomb Corp by Jackbox Games and moderating it, but I have a question about the recording. Since the game uses your phone/tablet/computer for the clues would I need to record both whatever is running the game AND my phone/tablet/computer?

As there's no runs on the board and you'd be the first, I'd say do what you feel is best and what you feel will ensure the fairest competition between potential runners.