You can either run the game on emulator anyway and not submit your runs to the leaderboard, or get a Wii/WiiU. Those are pretty much your only options.

Also, there's no need to ask the same question in two different places: https://www.speedrun.com/The_Site/thread/2wzsf

Just run the game using one of those consoles, there’s no difference in this regard since both can play a Wii game.

Dolphin is an inaccurate emulator, you’ll be hard pressed to find a Wii or GC game that lets you use it. It’s often quite faster than a real console so it’s not usually allowed since it’s unfair.

The best thing to do is to get a capture card to send the game feed directly to your computer. If you can’t afford one, simply record the game with a phone or tablet and upload the footage to your computer after the session.