Vigilantes regard the criminals and people they target as living outside the social bonds and communal ties that hold our society together.
The target must also be punished, and punished outside the law. Any and all legal matters on the subject are seen as unnecessary intrusions
Vigilantes do not care to wait for the police to finish their investigation, and they care less about any court's determination of proof. What they do care about is justice -- quick, final, cost-effective justice.
These are all romantic notions that feed an appetite for punishment more than an appetite for vengeance.
Punishment is the foundational matter of justice, and those who deserve punishment also deserve to pay (lex salica) or receive some kind of harm equal to the harm they have done (lex talionis).
That is the reason we have a system of laws and courts -- to sort out the particulars and differences between a criminal who deliberately commits a crime and one who accidentally commits a crime.
Nor is vengeance satisfying. Almost anyone who's ever thought about it knows than vengeance is an un-tempered emotion like fear, lust, and anger. Justice and punishment should NOT be guided by banal, primitive, un-tempered emotions.