First, anyone who thinks private law suits are the solution to policy problems just has been ignorant of what has been happening in the real of so-called juris-prudence in the last 20-30 years in the US.
Second, there is a legal technical issue that arises from this called standing. In order to bring suit you have to be able to demonstrate a substantial injury to yourself as an individual, not an abstract injury, but a real one. It is a fundamental principal of American jurisprudence, deriving from common law, a fundamental constitutional principal, and one that is not lightly cast aside because it means that lawsuits could explode as we file against each other for a host of imagined slights and injuries.
For instance, members of Congress generally lack standing to sue the executive branch over interpretation in enfrocing a statute. Knowing that the meatpacker 1000 miles away hires illegal aliens does not mean that I have a direct injury. I would have to show that I was denied employment as a consequence, or that I suffered food-poisoning as a direct result of using unlawful labor.
This is a simplistic and not well thought out solution to a law enforcement problem.
So it would be hard for an individual to win a civil case against an employer who cheats because it would be hard for the individual to establish legal standing. Would it be easier for a community (not the city or county government, but a group of individuals in a community) to file a class action suit against an employer who cheats?
Actually, Congress authorized private lawsuits for junk faxes and telemarketing calls. If a $500 lawsuit per junk fax stopped the junk faxes, then a $500 lawsuit per illegal alien hired will stop the hiring of illegal aliens.
I think Congress has the power to create private lawsuits over illegal immigration because it already has authorized RICO lawsuits on this subject - google “Howard Foster” and “RICO” for more info. It just needs to simplify the procedure to expedite the filing of more lawsuits.