I used to play GTA V on a Xbox and then sold my Xbox when I got a better PC, so now I play GTA on my PC with a controller... Can I speedrun with a controller?

You are allowed to speedrun with a controller and some runners use it for driving but shooting is still better with mouse and keyboard. If you use an Xbox controller it will just work, if you're using a PS4/PS5 controller you need to install a program called DS4Windows first.