You can put a claim in, but it will likely be denied as you rear-ended her; the law expects a reasonable driver to maintain his distance from other cars on the roadway. Don't be surprised if she doesn't begin a claim against you.
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As the following driver you had an obligation to not rear-end the vehicle, but, the lack of back lights could play a role in whether or not you are responsible or can claim that the other party was at fault. You would need to show that the lack of lights caused the collision and even then it would be an "uphill" battle.
Report it to your insurance company and they'll handle it. You both likely will be found partially at fault. Her license status had nothing to do with the collision. The brake lights are an issue but you still had the duty to come to a safe stop without hitting her. Good luck
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You can certainly put in a claim. Odds are that the respective insurance companies will find a degree of fault with the other driver as well as yourself. It certainly does not hurt to put in a claim with both insurance companies. That being said, the fact that you rear-ended the other driver does put you at a disadvantage. Good luck.