In some states where an ambulance is running through a red light for "in duty" and causes an accident, the ambulance company can not be held liable for negligence as state statutes provide immunity to the ambulance company and its employees because the ambulance company was acting on behalf of the government to protect its citizens.

Other than a gross negligence claim, what claim can be brought against the ambulance company for its negligence where its injured passengers are harmed from the ambulance's negligence?

Based on scientific literature, case law and my personal observations, ambulance have a high risk of crashes involvement.
With the growing ambulance industry, malpractice suits can be brought against the operator/ drivers of ambulance. To say, in today's context, malpractice are not just for doctors anymore.

I found case law showing that gross negligence, public policy, and acting outside the scope of duty are claims of action against the ambulance companies.

I was just surprised that laws created immunity for the ambulance drivers, employees, and the company.

most states enact statutes for emergency response personel and grant them a fair amount of immunity. they are considered a necessary service and the law tries to protect them from suit as much as reasonably possible. that's why the standard is "gross negligence" and not simple "negligence" in order to maintain a suit against a private ambulance service.