Playing Webkinz on Wii – Not the Main Game, But Two Spin-Offs Exist
You cannot play the main Webkinz World (browser/Flash/HTML5) on Wii. The Wii's Internet Channel (Opera) never supported Flash well, and the service ended 2020.
Wii Games Branded "Webkinz"
Webkinz World (Wii, 2009) – Mini-game collection (trivia, arcade, pet care) set in Webkinz universe. No plush code required. Single-player only.
Webkinz: Spooky Mansion (Wii, 2010) – Puzzle-adventure in a haunted house; 12 mini-games. Also standalone.