Are camera recordings on a Switch Lite allowed here as there is not a category for TOTK on Lite or should I request it for Switch Lite?

As long as the video and audio is clear on the recording, and all other submission rules are followed, it's okay to submit. Just leave Platform as default Switch on the submission page.

I'm new to speedrunning/video recording games.

Do you know of some of the best methods for recording a run? Particularly on a Switch family device?

You can either film the screen of your switch/TV or use a Capture card - I think there are no other possibiities for now since SysDVR Homebrew is banned for TotK.