In societies without a police system, citizens' justice was already pretty common. For example, in 14th century England, 'raising the hue and cry' was a way of gathering the locals to chase down criminals (or assemble a lynch mob, depending on whether you regard them through snobbish 21st century lenses or not.) Similarly, in Ancient Rome, even when the Vigiles were established under Nero, justice still involved private prosecutions, and very little police force (Lictors and Roman Knights seem to have occasionally been thrown in as security guards, but only very occasionally); so people naturally took matters into their own hands.
Less 'vigilante', more 'Right, everyone. My cloak has just been stolen. Cassia, my dear, get a dedication to the Gods ready. ((Many of the little stones dedicated to Gods in Roman religious sites-for example, at Bath-are citizens begging the gods to give them their damned cloak back, or at least call down a curse on whoever stole it.)) Slave Tertius! Get your cloak, my sons, and my sword. We've got a thief to catch. Slave Quintus, contact that lawyer fellow, you know the one-Cicero, I think his name is. Ready a court case-oh, and summon my clients, will you? Gaius Strabo served with Sulla out South, he ought to know a few tricks...'