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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.
Sounds like Taxi Driver. Travis Bickle works at night. Fantasizes about cleaning up the streets. He protects Jodie Foster's character. However The bullet proof coat doesn't fit.