Joyous news for all Wii-owning brewers, the Wii can run unsigned code without physical modification! The method, known as the "Twilight Hack", has been floating around the internet for a little while, so sorry if this is a little late. As the name implies, it requires a copy of "The Legend of Zelda: Twilight Princess", as well as an SD card and a computer that you can write to an SD card from. It works through a stack buffer overflow, and is the only way that you can run homebrew without a modchip. Note that GC applications won't run through this method. 
The hack works with the following disc versions:
 RVL-RZDP-0A-0 JPN – working
Some minor problems reported e.g. Some greeny lines around the top area of the screen that appear with the penguin and if the TV type is set to 480i/p (60Hz or EDTV mode) the bottom is cut off

RVL-RZDE-0A-0 JPN – working
RVL-RZDE-0A-0 USA – working
RVL-RZDE-0A-2 USA – working
RVL-RZDJ-0A-0 JPN – not supported yet

The latest available version of the hack can be downloaded at: http://chaoshq.de/~crediar/twilight-hack-v0.1-alpha3a.zip

As always, read any included documentation, ask questions when necessary, and do this at your own risk!

Cheers, all []D^_^