Dolphin is the only GC/Wii emulator I know of (if there are others I doubt they are as good as Dolphin).

And if you view the rules for the categories here (or try and submit a run) you’ll see that emulator is indeed banned for this game.

Correct, we do not accept runs done on emulator. Furthermore, we do not accept runs done on burnt discs, and we severely restrict what homebrew can be running during the run to homebrew that does not change timings in comparison to an unmodified console. See our discord for more information and if you have any questions, it is linked in the sidebar.