I don't know how good I am going to get at speedrunning, but I am very interesred in doing it. I think I might get a capture card at some point but don't know if it's worth it just yet, for starting out.

I know for super mario odyssey, using a camera/mobile to record the screen on video should be fine, but how common is it for other games? should I be fine with that temporary solution?

As long as the picture is reasonable without too much pixelation it should be fine. I know quite a few Wii runners use cameras due to not having capture cards. As long as nothing is obscured it should be all fine.