Yes, it can! But only by using a Generation 4 glitch where a transformed Pokemon that uses Rage and then kills their opponent with a non-Rage move keeps their stolen moveset permanently.
It’s possible to teach Machamp Mimic in Generation 3, so through some careful maneuvers it can transform and take advantage of this glitch.
Through all of this, it’s possible to teach it Fissure, Horn Drill, or Sheer Cold. I recommend going with Sheer Cold. It’s quite powerful in Generation 4!
The whole process can be seen here.