So I currently own a PSP Go, but not all games are available in the store, like Kao Challengers which I'd like to speedrun. So what would be the best PSP to get for speedrunning? Or is an emulator like PPSSPP just fine?

2000 and 3000 are the ones to go for I'm pretty sure. I know that the 1000 model doesn't have the video out feature. Depending on the game you're interested in, some may or may not allow PPSSPP. You'd have to check with the community behind the game on their ruling. The main issue is how much time emulator saves compared to UMD, which is countered by utilizing custom firmware and ISOs on the memory stick instead for console.

As Noishiru said, check with the community. I routed parts of Sword Art Online: Infinity Moment and on my PSP the game lags during some animations and the like. With PPSSPP however, there's no lag at all and barely any loading times, making it much, much faster, so I ended up not using emulator for the game at all when timing areas.

If you need official hardware Vita TV can run most purchased PSP games. Don't have much experience with this myself, but that's what I've seen romscout use for prinny 1.