I recently discovered this website and am very new to speedrunning. I'm not sure I'm ready to buy a capture card as most options are rather expensive, so I was curious if you are allowed to record runs with a camera or a phone? Thanks!

It is generally allowed, as long as the video has good enough quality, you can see the game screen clearly, and the audio is clear as well. You should check with the mods/community of the games you want to run, just to be sure. If you intend to speedrun browser/PC games, then that won't be necessary.