To best answer this question, we first must correct the terminology; but most briefly, the answer usually is “we do this because current efforts are insufficient, and the alternative is greater impunity for murder, rape, other assault, robbery, and other damage.” (and I of course ignore the case where this is a royalist telling you, the untermensch, to accept your servitude, lying back and taking it)
“The law” is a general term, for example it could mean the whim of a king, or the stipulations of a legal document approved by a popularly approved legislature. This response could get really long if we talked about every combination of these – but normally you talk about items of the basic just law – punishment for murders, rapes, other assaults, robberies, and other universal offenses – so that is the law we will talk about here. (And, talking about the concept of legal rights, not grounded in something like just law, the questioner could be referring to any structure, including slavery, in which the slaves have no rights to anything – so the question becomes trivialized.)
As an example of the just law, let us consider murder. Since this question begins with the assumption of the knowledge of rights (human or legal), then we need to clarify: it is a human right not to be murdered (not justly killed, for example if you are on a murder spree, you have no right to continued life). However, we know that there are people who decide to murder others, despite our assertion of and agreement on this human right. Then the question becomes: what do you do about it?
To simplify, there are two possible responses – you could rely on a professional police force to neutralize the threat, or you could kill the murderer yourself. It is odd to speak of a “right” to a professional police force, or a “right” to kill the murderer yourself, since circumstances sometimes dictate your response, and even in general, a professional police force is not the only appropriate means of law enforcement. If the murderer is an active shooter, you know more people are likely to get shot and die if you don’t kill the murderer yourself. However, if you believe the suspected murderer was purposeful and that purpose was limited (e.g. killing his wife), you may have plenty of time for an investigation, which would allow you to avoid killing the wrong person in certain situations where you would be misled. If you say an individual only has the “right” to allow a professional police force to operate, then you are denying the right of the future victims of the active shooter to live. If you say an individual in every situation has the “right” to kill whom they think is a murderer, regardless of how poorly the investigative process proceeded, then you are denying the right of the sometimes innocent accused to live.
Instead of using the term “right” in this instance (something given by God/nature/universal agreement/obvious or required logic), instead we should use a term like “prerogative” (if you mean there to be a very definite assignment on the basis of unique characteristic) or “license” (if it is somehow earned or gatekeeped) or “assignment” (if no one in particular, just someone has to do it), or in this case I currently suggest “imperative” as it carries the connotation of action as urgently needed. In a well functioning, well funded, non-corrupt justice system, the government would have the prerogative to investigate and hand down corresponding punishment in most situations, but an individual still has the imperative to kill the murderer in situations to which the government cannot respond as effectively.
Hence when the original question accuses “you don’t have the right” the response is, “both a good government’s prerogative, and an individual’s imperative, to enforce the law, both come from our right not to be murdered, raped, robbed, and otherwise assaulted. The situation dictates who does what.”
The second part of the question relates to whether anyone enjoys a prerogative, license, or similar limitation on law enforcement actions at a given time. Remember that the justification for assigning a government (that is, the individual human police officers, judges, etc.) the prerogative for law enforcement was to reduce the number of deaths and other misdirected punishments caused by incorrect judgments about who is guilty. However, if those individuals cannot reduce the overall number of deaths, for example because they do not have the investigative or punishment resources, or because they are corrupt and allowing criminals to go free and re-offend, then the prerogative (exclusive privilege) no longer more completely realizes the goal of upholding individuals’ rights to avoid murder, rape, robbery, and other assaults. Other strategies, such as vigilantism more completely realize this goal in this context, as allowing impunity will result in the greatest amount of crime.
Another argument often raised against vigilantism is that giving license to individuals to enforce law will lead to score-settling, organized crime, grudge escalation, and other sorts of criminal behavior going unpunished. These factors certainly are significant reasons arguing against vigilantism. A correct decision would somehow have us weigh the deaths of government prerogative vs. the deaths of vigilantism. However, it is practically impossible to estimate the deaths under one system vs. the other, as the transition period takes a significant amount of time, and you only live in one system at any given moment. Hence there is unavoidable ambiguity about the precise threshold, and since usually the problem is lack of resources (to fund a proper professional non-corrupt police force) you don’t have a good way to measure that.
However, there certainly is one threshold at which vigilantism clearly would improve the situation: when deployed against notorious criminals which the government cannot or does not eliminate. In this case, there is no concern about indiscriminate violence, except in the matter of collateral damage, which with lack of resources, practically cannot be addressed by either the government (if funding is the issue) or vigilantes.
Another situation is the holding of property, territory, or hostages by notorious criminals. Again, as notorious criminals/enforced territory, attacks and seizures against this are not indiscriminate.
Another situation is the protection of own territory with inspections/cameras and accounting for individuals and property. If transactions occur in an aboveboard manner and people are not being harmed, then the vigilantes cannot be committing serious just-law crimes of the sort that leads to preference of a sometimes-inept government.
These situations do not cover all matters of legal administration and punishment of all crimes in the most effective way, but if the just law routinely is violated with impunity, vigilantism is a clear improvement in these cases.
Another argument that is raised is the matter of alliances, and doing the bidding of rival armed criminals. If vigilantes take action against one set of notorious criminals, the others will benefit, and that might be strategically incorrect. The matter of correct alliances involves a strategy that results in the optimum balance of power/fewest just-law violations; often this is attacking the strongest armed groups in alliance and then fighting each other or smaller groups. Unfortunately the matter becomes rather complex when we talk about corrupt governments, for example. Is the corrupt government, half the time already allied with various notorious criminals, the most powerful criminal body? In many real-life cases, it is, and obviously war with it, or its criminally allied factions, is the default alliance orientation. In other cases, the government may be too weak, so the vigilantes strike out at the other groups that cause them the most damage. Is that an error? Not if the other notorious criminals leave that group alone, unless you needed that group to take some larger one preying on some other group of people. In short, the correct strategy by itself may not be at all obvious, and in some cases, the correct strategy is kill everyone that kills me.
So further to explain the matter of alliances, does the government even have the correct strategy? Does the government even have the prerogative to dictate strategy if the government is corrupt, or so weak that its own survival as an armed faction is threatened? Does the government have the intelligence resources to know the best strategy, if it is that weak? It is possible that no one can know, and oftentimes in poor countries none of the peasants feasibly can know, the best approach. Yet, notorious criminals still act, so some action must be taken against some of them, by some party, to prevent impunity. Hence, we are not in any position to criticize their actions on the basis of “right”; at best we can offer the tactical alliance advice to the extent we believe it is reliable. What we do know, and the vigilantes do know, is that acting out against criminals to their ability, is justified by the crimes they have committed. To assert that vigilantes do not have an imperative is to assert that the continued impunity is the best outcome; but we know that impunity is the worst outcome with respect to individuals’ rights not to be murdered.