Of course they're probably letting him do it, but vigilantism, especially at the scale he's performing it would be a felony. I wonder how many felonies Batman has done that the cops have turned a blind eye to.
Also, how fucking slow are the people of Gotham that the one guy that could possibly afford to be Batman is Bruce Wayne and nobody ever suspects him? There isn't one investigative journalist that tries to get a visual on Bruce Wayne while Batman's clapping poison ivy's cheeks.
That leads me to believe that not only is he into illegal vigilantism, but he probably has spent hundreds of millions of dollars in hush money to reporters, detectives, cops, et cetera all to keep his secret.
So he's bribing that many people and committing that many crimes every night even though it's in the name of justice. That makes Batman literally the most corrupt criminal in Gotham.