Correction: The Nintendo Wii does have built‑in Wi‑Fi (802.11b/g). You do not need a wired connection or USB adapter unless your Wi‑Fi is unavailable or unstable.
Nintendo discontinued the Nintendo Wi‑Fi Connection service on May 20, 2014 (affecting DS/Wii online play, leaderboards, matchmaking for most games). The Wii Shop Channel closed January 30, 2019.
For modern online Wii gaming, look into the Wiimmfi community project (requires homebrew).