So I have basically no gear for screen capturing, but I'd like to try speedrunning. Am I able to just, like, record it off my phone, and then post the video? Or does it depend on the mod?

Usually that’s fine, but check the rules for the game you’re planning to run.

The rule of thumb I’d say is as long as your recording device can see specific text or number on the screen (like coin count in SMO or star count in SM64) then it’d be pretty much guarantee that you’d be fine but certain games have different rules and some games may not even need video to be verified.