I really shouldn't have to explain this, because it's common knowledge, but you do not need a Homebrew'd Wii in order to activate cheat codes.
Someone created a corrupted Stage Builder stage which, when displayed on the Stage Builder menu, causes Brawl to suddenly cut out and load whatever boot.elf file is loaded into the proper directory of your SD Card at the time, such as Gecko OS. The hack is formerly known as Stack Smash, and it can be used on any Wii, modded or unmodded, and regardless of the version of the Wii.
There is a prerequisite to this, though, and it requires that you delete all other existing Stage Builder stages on your Wii, or else it causes the game to freeze.
In addition, you only need one SD Card to do this, because, once a code is loaded, the SD Card does not need to remain in the Wii for the codes to stay active. Codes such as "No Tripping" remain in effect even if the SD Card is not present in the Wii. The only one exception is the File Patch Code, but only because that code calls for Brawl to actively read files from your SD Card in order to work.
tl;dr - In order to run a "No Tripping" tourney, any Wii can be used, regardless of specs, and you only need one SD Card to do it.