What is the ruling of Islamic law regarding an employee filing lawsuits against owners of uninsured cars, knowing that insurance is mandatory in his country?

If a permissible cooperative Takaful insurance is available in your country, there is no objection to you filing a lawsuit against those who do not insure their cars. This is because the ruler's mandate for insurance was for the public good, and the actions of the ruler are tied to the public interest. "Coercion can be a right of the ruler, authorized by the Lawgiver, to repel injustice or to realize a public interest."