Playing Sonic Adventure on Wii: Via GameCube Backward Compatibility

No Sonic Adventure game was natively released for Wii. However, you can play them via GameCube backward compatibility on the original Wii (model RVL-001):

• Sonic Adventure DX: Director's Cut (GameCube, 2003) — Enhanced port of Dreamcast SA1. Chao Garden, Mission Mode, 60 FPS (mostly). Plays on Wii via GC slot.
• Sonic Adventure 2: Battle (GameCube, 2002) — Enhanced SA2. Chao Karate, 2P battle, all characters unlocked. Plays on Wii via GC slot.

Requirements

• Original Wii model RVL-001 (GameCube controller ports + memory card slots under top flap). RVL-101 (Family Edition) and Wii Mini lack GC hardware.
• GameCube disc (or backup via homebrew).
• GameCube controller (recommended) or Classic Controller.

Modern Ports (Better Experience)

• Sonic Adventure (DX) — Steam (2010), PS3/360 (2011, delisted), Xbox One/Series (back compat).
• Sonic Adventure 2 — Steam (2012), PS3/360 (2012, delisted), Xbox One/Series (back compat).

Best way today: Steam versions (widescreen, mods, 60 FPS locked). Wii GC backward compatibility is for purists with original hardware/discs.