I only have liability coverage and was involved in an accident and the other driver was at fault and cited at the scene. The driver was a cab driver driving a company taxi cab and is arguing he was not at fault and his insurance company is saying they are not sure if they are going to accept liability for my vehicle.
Will they still pay for my vehicle even if I only have liability coverage?
If they were at fault, they have to pay for your damages. Your liability insurance will only cover THEIR damages if you are found at fault.
Since they don't know if the are going to accept liability, you need to get your own insurance company involved to determine fault. They will do an independent investigation and then, if both carriers disagree, then both go to court on that issue. The arbitrator will decided.
The only way to get your damages paid for at this point is to found them liable. You don't have first party coverage, so that will be difficult to show.