Can a lien holder file an ins claim on customer's auto policy?

No a lien holder can not file a claim against the insurance company as they are not the named insured, you are. Although if there is a lien on the vehicle the insurance payment for damages to your vehicle will be in your name and the lien holder name.

They then might require that you fix the vehicle so they can protect their interest in the vehicle. each lien holder is different.

CORRECTION: If the lien holder is named on the policy and the vehicle has been repossessed, the lienholder has a right to recovery under that policy.