no, the full channel is working only on Wii (the wad has not been updated).
We prepared a version with Wii/vWii choice, but I didn't release it because the full channel is not updated anymore since r1180.
it lacks a lot of features (no gamecube USB support) and bugfixes.
Also, installing the "unofficially patched" Wad version for vWii would have ending with a bug due to auto-update. (I know some users thought it was intelligent to convert the full wad to vWii and release that bad version...)
the URL in the existing full channel is still the Wii version, releasing a new version would automatically replace the vWii wad and users would complain it broke their console.
I recommend to only install the forwarder (even on Wii) and use the automatic installer only to get/update the dol on SD.
Edit:
I see Flash_drive asked this to get a "nice channel installed on the menu", but don't fear ! the forwarder is identical
you'll have a channel installed on the menu.
the only difference is that it will require the boot.dol on SD or USB, but you can't use USBLoaderGX without SD nor USB anyway, so you'll need an external device EVERY TIME. just put the dol on it. you won't see a difference, except that it's easier to update
also, use THIS fat32 formater tool (everyone else are already suggesting this one)
http://www.ridgecrop.demon.co.uk/index.htm?guiformat.htm