I've never played (Ultra) Sun/Moon, so I don't know for sure if there's no HM moves in the games, but I can't find any. I believe that is the reason.
In Generation 6 (X, Y, Omega Ruby, Alpha Sapphire), there ARE HMs, and Greninja can learn some of them: Cut, Strength, Waterfall, and Surf.
Source: PokemonDB's Greninja moves list.
However, on the move list for Greninja in Sun/Moon, cut does not appear at all. Since Ash-Greninja is Generation 7 exclusive, cut is not possible for him to learn in-game.
Because of this, It cannot learn cut on Showdown!.
This is the reason that makes the most sense. However, even that leaves something to be desired.
Like I said, I haven't played Sun/Moon, but I think that it is possible to move your Pokemon from the Gen 6 games to the Gen 7 games. If this is possible, my idea should be thrown out. If not, then my reason is probably right.