Wii Hardware/BCM4318
The Wii and Wii U features a Wi-Fi chipset from Broadcom, specifically the Broadcom 4318.
This Wi-Fi adapter is unfortunately heavily constrained due to it's connection, it's connected over Sonics Silicon Backplace (SSB), connected over SDIO on OHCI1 port 1, on the Hollywood/Latte chipset. The overhead of interfacing with this Wi-Fi card along with various standard network protocols are very taxing for the Wii's CPU.
Compatibility
Wii-Linux has included support for the Wi-Fi chipset since at least MIKEp5[check]. Upstream Linux also has drivers for all components of the Wi-Fi stack available, although they don't seem to function on the Wii. It's unsure if this is known about by the upstream maintainers, but it seems to have been broken since at least Linux 5.0.