I remember watching this movie a long time ago, sometime in the 90s.
All I can remember clearly about it is that this guy is an insomniac and never sleeps but at night he goes out and fights crime in some way or form. I clearly remember something about a trenchcoat he wears that is bullet proof and he used it to protect a child at one point. He also swapped it for a new one at a shop in one scene.
"The Owl" from 1991 stars Adrian Paul as an insomniac vigilante wearing a kevlar trench coat. Basically an unpicked TV pilot, it was also called "Night Owl". From IMDb:
Alex L'Hiboux, a ruthless mercenary-cum-vigilante, is known as 'the Owl' because he never sleeps. His insomnia comes from a combination of a medical disorder and recurring nightmares of the murder of his wife and daughter. Alex is approached by Lisa, a young girl whose father is missing. She awakens painful memories of his own child, but after some persuasion from a policewoman friend, he agrees to help her.