Sometimes people feel that they want to stop criminals because they feel that they can’t wait until the police come to find the criminals before the criminals ran away. It is not recommended that a vigilante harm anyone because in the United States, every criminal must be allowed to go through the judicial process. Everyone is considered innocent no matter what before going to court.
So, in the United States a vigilante can’t shoot and kill a serial killer without going to jail themselves. They could only do it if they felt their life was threatened and they would have to prove it. In other countries, it may be different to shoot and kill a known serial killer. Most discourage it.