Vigilantes are individuals who band together for the purpose of bringing criminals to justice because they deem the legal authorities as inadequate to deal with criminals.
First Romans 13:1-5.
"Romans 13 Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God. 2 Consequently, whoever rebels against the authority is rebelling against what God has instituted, and those who do so will bring judgment on themselves. 3 For rulers hold no terror for those who do right, but for those who do wrong. Do you want to be free from fear of the one in authority? Then do what is right and you will be commended. 4 For the one in authority is God’s servant for your good. But if you do wrong, be afraid, for rulers do not bear the sword for no reason. They are God’s servants, agents of wrath to bring punishment on the wrongdoer. 5 Therefore, it is necessary to submit to the authorities, not only because of possible punishment but also as a matter of conscience.
6 This is also why you pay taxes, for the authorities are God’s servants, who give their full time to governing. 7 Give to everyone what you owe them: If you owe taxes, pay taxes; if revenue, then revenue; if respect, then respect; if honor, then honor."
And 1 Peter 2:13-17
"13 Submit yourselves for the Lord’s sake to every human authority: whether to the emperor, as the supreme authority, 14 or to governors, who are sent by him to punish those who do wrong and to commend those who do right. 15 For it is God’s will that by doing good you should silence the ignorant talk of foolish people. 16 Live as free people, but do not use your freedom as a cover-up for evil; live as God’s slaves. 17 Show proper respect to everyone, love the family of believers, fear God, honor the emperor."
Both these apostles, in accordance with Jesus' teaching exhorted obedience to governments, however the only time they should be disobeyed, which can be in the TANAKH, is if they tell you to do something against God's commands. Such an example is in Daniel 2 when Daniel's friends Shadrach, Meshach and Abednego are thrown into the fiery furnace after refusing to bow to an idol created by Nebuchanezzar and of course the story of Daniel in the lion's den that needs no introduction.
Vigilantes are outlaws, doing things that they shouldn't be doing, namely taking justice into their own hands, and self defence is NOT the same thing as being a vigilante.
It is the right to the government to dispense justice and punish the guilty for that is what governments are for. It is not for the ordinary citizens to do as that has not been granted to them.
Those who are familiar with Batman should have a semblance of an idea that he, though in league with Commissioner Gordon, gets into trouble with the police for his actions. One prime example from Batman can be given can specifically be given from Batman Begins, the scene were he has to get Rachel an antidote.
Although I am a huge fan of the Dark Knight Trilogy, I do not condone vigilantism nor would I encourage Christians to become a vigilante, nor would I want to become one.
There is no biblical nor exegetical basis for Christians to take up the life of a vigilante.