Worth noting that the usual concerns of vigilantism don't actually apply to Batman,

• He works directly with the police, and even then specifically with the cops that have passed the vibe check and can be trusted

• He mostly avoids direct confrontation unless he's been attacked or someone is actively in danger, and even then, what genius is seriously gonna willingly tango with the goddamned batman?

• Basically everything else he does save for regularly committing cyber crimes to steal Amanda Waller's AP class homework for his own cases or just to keep tabs on the feds given how much shit they get up to in DC comics, would be protected by Good Samaritan law, basically you get special legal protection even if you committed the crime beforehand if you drop what you're doing to save someone's life. Most often this is used to protect college kids who bring in their ODed friends to the hospital from being prosecuted for underage drinking or drug charges or standers by who intervene to rescue car crash patients from liability suit, but DC almost certainly has a white list of recognized vigilantes who get this protection for rescuing civilians and taking on police work on volunteer basis.