There are two protagonists.
The first is clearly Marion Crane, who we invest our emotional energy in as she attempts to steal the money and start a new life with her boyfriend Sam Loomis. Then… after a shower that goes very very wrong…
The second protagonist takes over and drives the narrative. That protagonist is Norman Bates, and it is one of the great shifts ever in film. We follow Norman as he cleans up, hides the body, and continues to try and "cover" for his mad mother. We were already sympathetic to him after his conversation with Marion in his office and the audience continues to "root" for him up until the end.
Since the "spoiler" ending is now common knowledge, the film's denouement doesn't have the power it did upon the films release – Hitchcock and writer Joseph Stefano created a film where the audience both cared for a thief and a psychotic murder. An incredible feat, groundbreaking at the time.
(A better question may be "who is the protagonist for Star Wars Episode I: The Phantom Menace?" Because there's not really one in that movie…)