It's just an expression. Lawyers don't literally chase ambulances. It's illegal for lawyers to accost people in the emergency room offering their services, and it's probably illegal to literally chase an ambulance as well due to traffic laws.
The story has more to do with suing someone for the personal injury that landed the patient in the hospital in the first place, but it's meant to refer to lawyers going out of their way to recruit clients who might not have even wanted to file suit in the first place, or to unethical, enterprising lawyers in general.
One of my buddies in LA got involved a multi-car accident. Within minutes he had a lawyer stop by his car and drop off his business card in case he could be "of assistance". So yeah, "ambulance chaser" an expression, but sadly very close to reality in some cases...
The story has more to do with suing someone for the personal injury that landed the patient in the hospital in the first place, but it's meant to refer to lawyers going out of their way to recruit clients who might not have even wanted to file suit in the first place, or to unethical, enterprising lawyers in general.