So I have a copy of the rarest game on 3DS Barbie Groom and Glam Pups and I see there isn't a speedrun record for it, I figured I should do a speedrun as whatever I get will be the world record by default. The issue is to my knowledge though that this isn't a game you can "win"

So how would I be able to speedrun it? Also how do you do speedruns of DS games as i've never done any before. Would love to get some help and try and get this game documented as well since there isn't 3DS footage online except one played terribly on an emulator that doesn't really count for anything.

You'll have to request the games you want on the site's thread. Make sure to read the rules fully and don't set your expectations high--the way you describe the game here, there's a good chance it'll be rejected. Unless you want to purchase a 3DS capture card or use homebrew, I'd just point a camera at the screen.