You can no longer connect a Wii to Wi-Fi for online services. Nintendo discontinued the Wii Shop Channel and all Wii/DS online infrastructure on January 31, 2019.
Why It's Gone
End of lifecycle: Focus shifted to Wii U and Switch.
Security: Aging protocols (SSL/TLS versions) vulnerable to exploits.
Cost: Maintaining authentication, matchmaking, and shop servers for a 2006 console wasn't financially viable.
Homebrew Channel (if installed) — runs emulators, media players, homebrew games offline.
Alternatives for Online Play
Wii U / Switch: Native online for modern titles.
Dolphin Emulator (PC): Netplay allows online multiplayer for many Wii/GameCube games (requires owning the games).
Wiimmfi (Community Server): A fan-run replacement for Nintendo Wi-Fi Connection. Works with Mario Kart Wii, Super Smash Bros. Brawl, Pokémon Battle Revolution, and others. Requires homebrew (Letterbomb) and patching game ISOs.
Your Wii isn't dead—it's just offline. Offline gaming remains fully functional.