Im about to try my first speedrun, but my computer is too crappy to handle streaming and playing at the same time without significantly slowing my computer, so is it possible for me to just make a screen recording instead?

By "screen recording" you mean record a video of you playing to a local file on the computer? If so, then yes. Many runners do that - record to a video file and then upload it, mostly to YouTube. It's also recommended to do even while streaming (if your computer can handle it), for backup in case the stream fails.

If by "screen recording" you meant showing a photo of the final screen of the game, than most probably it will not be allowed. Unless the rules of the game permit it (and in most cases they should not), all runs submissions require video proof.

The middle-man to what @Oreo321 mentioned is putting a camera aimed on your monitor (phone, actual camcorder) and upload that to youtube. So, a recording of your monitor where you speedrun the game. That should in most cases also be A-ok, as long as the audio and video quality are passable.