How can Aoshi Shinimori walk/run/fight when he got shot in both legs? In the fight with Kanryu Takeda, Aoshi Shinimori gets shot in both legs, in what appears to be above the kneecaps. Yet come a few months later, he is able to fight perfectly fine in the Kyoto arc, despite these injuries. Does Nobuhiro Watsuki ever explain this at all? Or is it different in the manga in some way, where he doesn't get as grevious of injuries? I'm looking for either an official explanation by the author or by the source.

TLDR: How can Aoshi Shinimori walk/fight with gunshot wound injuries?
While bullet wounds can be very dangerous, according to this Wired article by a combat medic, they can also be relatively trivial. The author retells the story of a man shot in the chest and neck six times who remained fully alert and responsive, and survived. According to Wikipedia, "depending on the extent of injury, management can range from superficial wound care to limb amputation." A bullet to the leg could shatter your femur, or it could cause so little damage a doctor would just slap a bandage on it (assuming the bullet passed through).

Moreover, a man like Aoshi would likely have trained in martial arts pain management techniques. Aoshi may well feel pain from his injuries during the Kyoto arc but could ignore it.