@Kemp and Rooster
There are many reasons a government may fail to enforce the law. For instance in the parking ticket example, the government decides that enforcing the law in all instances is inefficient. It makes more sense for the government enforce the law sporadically. It's an effective deterrent, and doesn't require the hiring of thousands of bylaw officers to patrol the streets. It serve the government's purpose.
In murder cases the government almost always chooses to enforce the law.
So why is the government failing to enforce the law? What do you mean by failure? Is the government unable to enforce the law. Are they able to enforce the law, but are choosing not to enforce the law in all instances? Do they have the power to only sporadically enforce the law? Are they ignoring crime in specific areas (e.g. ghettos)? Has the government failed to enforce the law because the perpetrator has gotten off on a technicality or a lack of evidence? Did the perp make a plea bargain to testify against a bigger fish in exchange for immunity? Is the crime so minor, that the government is choosing not to pursue it? (As an example of the last one, some kids decided to toss some eggs at my dad's house. Now that's against the law. The police didn't care much though. They didn't go through hell and high water to find the bugger who did it.)
What constitutes a failure by the government to enforce the law?
Without being clear on that, it's hard to create an argument, because depending on what was intended by the idea of the "government failing to enforce the law", the applicable arguments change.
Although maybe that's too complex a thought for the level of debate. If it's only high school or whatever, you can just pick one and run with it (unless the side arguing for vigilantism is required to define the terms).
"ok but vigilantes never form over a parking ticket. Vigilantes form over major crimes. "
Vigilantism can occur over slight legal infractions (e.g. someone running a stop sign or a red light, tailgating...yay for road rage!).
Or...Billy egged my car. I'm going to shoot him. Then you have fathers who rape the men who raped their daughters. Vigilantism relates to a broad spectrum of activity, from minor incidents to major incidents. This is because the motivation for vigilantism is typically vengeance. And people will take revenge over small legal slights as well as large ones.
Here are the stories for the last two I mentioned. They both occurred in 2008 I believe...quite tragic really.
Man Rapes Stepson for Raping Daughter
A 32-year-old Fort Worth, Texas, man called police and had his 18-year-old stepson arrested for aggravated sexual assault against the man's 8-year-old daughter. The 8-year-old had been anally raped.
Against the man's warning, the man's wife posted bail for the stepson. However, when the stepson called for a ride home, the man took the call. But, after the man picked up the stepson, he didn't drive straight home.
Instead, he took him to an abandoned house, assaulted him with a baseball bat, and anally raped him with a wrench. The man turned himself in for aggravated sexual assault. The man does have a criminal history which includes other assault charges.
Boy, 16, killed after egging cars
Sunday, August 31, 2008 3:40 AM
By Suzanne Hoholik
THE COLUMBUS DISPATCH
ADAM ALEXANDERDispatch
Georden Burton is comforted by Westland guidance counselor Christi Shore at the football game last night.
Westland High School junior Garrett P. Burton, 16, was shot and killed near Hilliard-Rome Road.
First, a moment of silence. Then the Westland High School band played the school's alma mater in memory of Garrett P. Burton at last night's football game.
The 16-year-old junior was shot and killed early yesterday, apparently in retaliation for throwing eggs at vehicles driving on Hilliard-Rome Road. Columbus police said Garrett died at the scene just before 3 a.m.
Garrett and at least one other boy were in the backyard of a house at 5402 Ripplemead Court when someone shot him in the head.
http://www.dispatch.com/live/content/local_news/stories/2008/08/31/TEENKILLED.ART_ART_08-31-08_A1_RPB6HC5.html?sid=101
Road rage cases are fairly well documented. You can find some yourself that conform to vigilantism. 
The point is that claiming people only take the law into their own hands when serious breaches occur...is wrong.
To serve is to survive