FLOSSMOOR, Ill.—Five years ago, Luke Lambert and his wife bought a red brick house on a tree-lined street in this Chicago suburb. Two years later, he bought a blue 1966 Ford F-100 pickup with plans to fix it up.

Then he got a phone call from the police: Pickups can’t park in Flossmoor’s residential areas unless garaged or being unloaded. His garage wasn’t big enough, so he parked at his wife’s grandmother’s about 10 miles away.

Now...