The vWii (virtual Wii) is the part of the Wii U's system for Wii mode. None of the System Menu exploits work in vWii, due to the vWii running a special version of the system menu, known as System Menu 4.3-U. Additionally, BootMii and Priiloader are not supported, although there are tools to recover from a vWii brick that can be used from Wii U Mode. Many different homebrew applications will not work on vWii, or will result in a brick if used.
While homebrew can easily be run in the vWii, there are not many ways to exploit Wii U Mode from the vWii. This is due to the fact that there is a clear-only register that determines whether the vWii sandbox should be enabled. Nintendo solved this in the Wii U Menu channel by making it simply restart the console. However, there are a couple of bugs, such as dumping the Wii U's boot0 and the HRESET hack, but none are very reliable for gaining code execution in Wii U Mode.
With an updated System Menu, LetterBomb does not work on the vWii. Additionally, str2hax does not work, because of the removal of the system settings menu. The same savegame exploits that worked on System Menu 4.3 also work here, although this is not necessary, as a vWii-only exploit known as Wuphax exists. Alternatively, Compat Installer can be used.