Hi so I don't have a capture card to film my speedrun so is it allowed for me to use my phone camera and record it there and upload it to YouTube?

Hi, I dont see any problem on that, if the video is good enough to understand what is going on and have a decent quality, they probably will accept your run! But you speedrun on Switch, PS4 or Xbox One?

Yea, I recorded runs on my phone when I 1st did runs and they all were accepted, so as long as the screen isn't too bright, dark or blurry for anything to be seen, you should be good.