I don't own a GameCube (and it would be too expensive to get one.) Can I upload a run with Dolphin if I promise I won't use any emulator cheats?

If emulator cheats aren't the issue, then I won't do the thing with the issue.

I think the reason emulator isn't allowed is because of inconsistent saving and loading times from the emu memory card, the easiest way to run is on a Wii with backwards compatibility (pretty common), an official GC memory card (59 and 251 are the slowest, and 1019 is the fastest), and then either a disc of the game or homebrewing the Wii and using a homebrew application called Nintendont that loads GameCube games from a USB.