Am I allowed to speedrun randomized pokemon games or is it that you have to speedrun normal games and can't be randomized?

You can speedrun whatever you want, however you want.

A better question would be: is there a leaderboard for such a thing?

I've played randomizers before and I think speedrunning it is pretty silly, but people speedrun games with just as much RNG so w/e

Some randomizers have been run in the past using a set seed (like OOT Beta Quest) however those categories are usually put on Category Extension/Meme boards because they are mostly silly and not taken too seriously.

Omg if that was a thing though I would so do it like the red and blue you can get the randomizer that adds the other gens pokemon to the game, and as long as everyone used the same randomizer and the same settings could be super fun. I love randomzing pokemon games, would always be fresh :) haha yeah I would love it but sadly it isn't a category. :)

You can still run it regardless of a category existing.

People do on SRL all the time.

I know :D I see it all the time like with hey you pikachu and stuff ^^ I may do it then that might make more people want to do it never know :)