Yes, the Wii U has built-in Wi-Fi (802.11n, 2.4GHz only). No external adapter needed.
USB-to-Ethernet adapter (official Nintendo or generic) plugs into front/back USB port. More stable for downloads/online play.
Nintendo Network (Wii U/3DS online) remains active for now, but eShop purchases ended March 2023. Redownload owned titles via "Redownload" in eShop menu.