Let's start this by referencing box art nobody wants to remember.
Our vaguely-Indo-Chinese billowy-armed Blue Bomber with frog legs appears left handed, as he's holding the weapon (a pistol, not the Buster Cannon) in his left hand. The pistol itself seems to be a traditional firearm, not an energy-based weapon. We also know that the arm cannon was something Dr. Light developed, using Solar shots (the many iterations of the Buster Cannon). This suggests the pistol image below references a time before Light made the cannon, indicating handedness. The Buster weapon itself is referenced as ambidextrous, but not the robot; this is similar to how a baseball bat is ambidextrous in nature, but handedness determines the swing. The pre-Buster pistol in the image is certainly not such a weapon, as while held in the left hand we have to infer that shells would eject out of the unseen side of the pistol (as no slide or ejection port is visible). This would make the gun itself a right-handed weapon, held by a left-handed person.
Following the art on the same page, we can see that standalone images of the various Mega Man incarnations begin to show the Buster on the left arm repeatedly. One could arguably say that the more plot in your Mega Man, the more right-handed he gets. There's a trope involved here; A Sinister Clue. For literally centuries, entertainment has placed the villain or the goofball as being left-handed. Even in modern movies, while the "main character" might be left-handed, it's actually a clue that the one character who is right-handed is the real hero of the story. This actually fits in with Mega Man's Lore, as from the Wikipedia entry, Mega Man was originally a battle robot made by BOTH Dr.'s Wily and Light. It goes on to say that Mega Man (originally named "Rock") was refitted after being a villainous battle robot:
"Following treachery by Dr. Wily in which he reprogrammed all of Dr
Light's robots, Rock was converted into a battle robot to defend the
world from Wily's violent robotic threats. Thus he becomes Mega Man."
This suggests that while the weapons he uses are ambidextrous, Mega Man actually started out left-handed and literally had his handedness changed for story purposes to match with Western culture's expectations and tropes. All because ugly-halfway-racist-proto-MegaMan pictured in the box art would have held a right-handed weapon in the right hand if he were ambidextrous or right handed, as the ejecting shells across the chest of a traditional firearm (as pictured) would risk damage and not fit with machine logic. He technically was born as a villain, so we see him with a gun in his left hand before Light ever modified him.
Edit: Captain N's Megaman doesn't count. Green with wrist launchers and toddler-sized. No.