If you only have liability insurance then you are unable to make a first party claim with you own insurance company since you would need collision coverage to cover your vehicle in an accident.

In most states if you have been hit by another vehicle that is found at fault in the accident you can make a claim through the at-fault party's property damage liability coverage. You thus would make a third party claim with the insurance company of the at-fault person that caused you damages in the accident.

If the other party was uninsured then you would either pay for your own damages or request the at fault party pays since he or she was at fault for the accident. If the person does not pay you, or repay you if you first pay for the repairs yourself, you normally can take the person to court and seek a judgment against the other party for the cost of repairs, rental car for the time the car was in the shop being repaired, etc.

If you are unsure of your state's laws your insurance agent should be able to advise you on how to make a claim with the other insurer but your insurance company itself cannot make a claim for you or help you get the other party to pay for the repairs since you only have liability coverages with them and not physical damage coverages such as collision and comprehensive.

You can also contact your state's insurance regulator for consumer advice on what steps to take regarding your situation. Many have online guides on what to do after an accident and how to make claims. If you are unsure of your state's insurance laws then this state agency should be able to help you understand the laws regarding your situation.