How can you get someone's auto insurance information if they are at fault in an accident and refuse to give it to you?

Assuming there was no police report filed, you will have to do the work yourself. Get a lawyer and an investigator and file a lawsuit.

Provide the information to your insurance carrier. They can normally get the insurance information from the State DMV using the license plate number of the vehicle that hit you.

Keep the license plate number and call the police. The police have all the information. They can find out the information of that car instantly, given the plate number.