As stated in the question, according to Wikipedia it the group in the film are a mockery of the American Nazi party, but does the influence the Nazis appear to have in Chicago, have any basis in reality? Is it just an excuse to have easily dislikable villains?
The National Socialist Party of America had their headquarters in a Chicago neighborhood called Marquette Park starting in the 70's. Even before then, Marquette Park was known to be an almost all white area where there were a number of racial charged and counter rallies. Martin Luther King tried to march through Marquette Park in the 60's and later said that "I have seen many demonstrations in the South, but I have never seen anything so hostile and so hateful as I've seen here today,".
But the court case they are referring to is almost certainly National Socialist Party of America v. Village of Skokie, decided by the United States Supreme Court in 1977. The Nazis applied for a permit to march through the Skokie, which was a largely Jewish suburb of Chicago. One in six of the Jewish citizens of Skokie was a Holocaust survivor or the relative of a Holocaust survivor.
Skokie was able to get an injunction against the Nazis preventing the march, but the ACLU sent a Jewish lawyer to represent them and after losing at lower courts ultimately prevailed at the US Supreme Court.
The Nazis did not march through Skokie, but did continue to have rallies at Marquette Park and fought another legal battle to have a Rally in 1978 that attracted over 2,000 people both pro and anti and 72 people were arrested.
Frank Colin, the Nazi leader said that he had used the threat of the Skokie march to win the right to rally in Marquette Park.
Conflicts continued in Marquette Park into the 80's such as when Crusaders for Justice attempted to march into the area and were kept out by police.
A great source on the legal case is When the Nazis Came to Skokie Freedom for Speech We Hate by Philippa Strum.