Do I have to Record my run with a capture card, or can I just record the screen with a camera.
And if I can use a camera, do I have to record the sound also?

"Refer to this thread for the leaderboard verification process: https://www.​speedrun.​com/smw/thread/f7cr2
Also, if you have any further questions the best place to ask is in Discord https://discordapp.​com/invite/0SkVJ6hE2KFkKlak" - Bramz

In short, using a camera is fine. Please record sound

Ok, but is it fine if the sound isn't perfect?
Or does that matter

It doesn't need to be great quality. Depending on what your time is you could get away without video at all, although if you can record I'd recommend it.