Sad news on March 27th. Goodbye, 3DS and Wii U eShops.

Does the Homebrew Browser support the Wii U's Wii Mode (vWii)?

The Homebrew Browser itself does support the Wii U's Wii Mode (vWii).

However, some applications, specifically those who write to NAND (Wii internal memory) or install cIOSs, might not work properly, or even be harmful, when running in vWii. Before running any application that you suspect writes to NAND, make sure it supports the Wii U.