Can an insurance company request the state to suspend your drivers license?

No, not usually. The insurance company has nothing to do with suspension of drivers licenses, that's up to the Department of Motor Vehicles or whatever it is in your state. I have seen times where a person owes an insurance company money from hitting an insured vehicle when the party At Fault did not have insurance required by law. In these cases I have seen our state suspend a drivers license until the balance is paid in full to the insurance company. This is because it was due to failure to obey a state law of not driving without minimum required insurance limits. I saw once where a 15 year old could not get a license until this was paid off in full.