This question reminded me of a similar short story that I believe I read in one of the various SciFi magazines, IASFM possibly, since I had a large collection of those, most from the 80's. It is definitely not "Why Johnny Can't Speed" although it is very similar.
It's set marginally in the future, or a semi-dystopian "future now" relative to when it was written. A man and his family are driving along the highway in a very standard family
car hovercraft when someone in an obviously high-powered vehicle becomes aggressive towards him. (He is cut off, tailgated, something, I forget exactly what.) There is a sequence of detailed road combat - with the man and his family increasingly terrified at the situation - until the man finally manages to take the punk and his hovercraft out by throwing his and his wife's safety helmets out of the vehicle so that they get sucked under his adversary's hover skirt. He is then horrified to realize that he is now on the top-10 list of road warriors, having just picked off one of the highest-ranked members and thus earning his position. His car console lights up with messages of challenges from many other lower-tier warriors as the map shows them starting to converge on his location.
It differs from "Why Johnny Can't Speed" in that it's accidental (well, it's his fault, but he didn't realize what he was doing) that the man gets involved in the car wars situation, he distinctly does not have a combat vehicle, and there was no family death involved.
edited to change some details as indicated by comments