If we were hit from behind last night but the driver has no insurance and our insurance has lapsed, what can we do?

Since the at-fault driver did not have insurance and your insurance lapsed, you will need to sue the at-fault driver/registered owner of the vehicle for negligence to compensate you for property damage (cost of repairs) to your car.  Depending on the amount of damage, you may be able to file your lawsuit in Small Claims Court.

You will need to mitigate (minimize) damages by selecting an auto body repair shop whose charges are comparable to other auto body repair shops in the area.  If you were to select the most expensive repair shop you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.  If you need a rental car while your car is being repaired, you will also need to mitigate damages by selecting a rental car with a reasonable rate.  If you were to select the most expensive rental car you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.  When you get a court judgment against the at-fault driver/registered owner, it would be advisable to pursue a wage garnishment to satisfy the judgment in case the at-fault party does not pay.

If you were injured in the accident, don't file your lawsuit in Small Claims Court.  When you complete your medical treatment and are released by the doctor or are declared by the doctor to be permanent and stationary which means having reached a point in your medical treatment where no further improvement is anticipated, obtain your medical bills, medical reports and documentation of any wage loss.  Compensation for the medical bills is straight reimbursement.  Compensation for pain and suffering is an amount in addition to the medical bills based on the nature and extent of your injury as documented in the medical reports.  Compensation for wage loss is straight reimbursement.  If  you were injured in the accident, your lawsuit  for negligence against the at-fault party should be filed in Superior Court.  Your state may have a different name for Superior Court.

You file one lawsuit for negligence against the at-fault party in which your damages (the amount you are seeking to recover) would include your property damage and personal injury claim. After obtaining a judgment against the at-fault party, pursue a wage garnishment.