Playing as Your Mii Online in Mario Kart Wii? Not Possible.

You cannot race as your Mii in Mario Kart Wii online.

Why

• Mii Integration Was Limited: MKWii (2008) predates full Mii network integration. Miis exist only in local modes (Grand Prix, VS, Time Trial).
• Online Profile = License: Nintendo Wi-Fi Connection identified players by Friend Code + License (Mii face + name), but the in-game character was always a standard driver (Mario, Luigi, Funky Kong, etc.).
• No Mii Character Slot: The character select roster has no "Mii" option.

What You Could Do With Miis on MKWii

• License Card: Your Mii appears on your license (seen by rivals in VS races, leaderboards).
• WiiConnect24 / Mii Channel: Send/receive Miis to friends' Mii Parade. Not in-game.
• Ghost Data: Time Trial ghosts show your Mii face.

First Game With Playable Online Miis

• Mario Kart 7 (3DS, 2011): Mii is a selectable weight class (Light/Medium/Heavy).
• Mario Kart 8 / Deluxe (Wii U / Switch): Full Mii support with custom outfits (Deluxe).

On original Wii, you're stuck with the 24 preset characters online.