The meaning of "delinquent" varies. In some cases, it's close to what you'd call a "problem child" or "troublemaker" in the US - just somebody that doesn't follow all the rules and doesn't flow with the crowd. Conformity is a big deal in Japanese culture so if you refuse to conform, you're often considered a delinquent.
Sometimes gyarus are called delinquents for the reasons above, but they're really their own kind of thing.
In other cases, it's more like "punk" is used in the US - referring to a specific subculture with its own set of norms. Japan had something like that called "Yankī" going back to the 70s. It mostly died out in the 90s and 2000s. There are a lot of manga about this subculture. It's not really my thing, but I enjoyed Angel Densetsu and Cromartie High School.
