So, of course, the Gamecube version is the ideal version of Sonic Heroes to speedrun on but I don't personally own a Gamecube console and I tried my attempts by playing the Gamecube version on the Wii.

Does playing the game on the Wii affect runs or does it not matter playing on either the Wii or Gamecube?

The only difference is faster loads depending on what version you have. However, in most categories we use in game time provided by the file you use in your runs, which means in any different console or emulator nothing changes.

Load differences only affect categories timed using real time, like super hard mode.

There are more significant differences in other console versions (PS2, xbox), but GC version on Wii is the exact same except for what I mentioned.