Before diving into the answer to that question, a little clarification is in order. For purposes of this posting, the question assumes that the accident was vehicle vs. vehicle, not vehicle vs. pedestrian. It also assumes that the drivers were of legal age and the children were properly restrained. If either of the aforementioned conditions were not met, a whole different set of issues arises.
A critical piece of any car accident is to determine fault and liability. If you are the driver and you are at fault (say you failed to stop at a stop sign, missed a red light, or didn't check your blind spot when merging and merged right into an oncoming car), all claims are going to be processed through your insurance company. If the other driver was at fault, it will be up to that driver's insurance company to help pay the bills. A police report can be extremely handy in proving fault.
Even children who are properly secured in a correctly installed car seat or safely buckled in a seat belt can suffer serious injuries in a car crash. Children can suffer from head injuries, fractured bones and even die. If the injuries are sufficiently severe, an insurance company may not be too willing to adequately provide for the child's care. Even if the child suffered minor or no injuries, the car seat may need to be replaced. The NHTSA has set forth guidelines that state when a car seat should be replaced after an accident. Many insurance companies use these guidelines to help them determine whether the purchase of a new seat is warranted.
If the insurance company is only willing to pay a minimum sum, or is refusing to pay altogether, you may wish to consult a lawyer. A car accident can bring a myriad of problems and details that need to be sorted out. If your child has been injured, the last thing you want to do is argue with an insurance company about what they will and won't pay. Personal injury attorneys have a lot of experience negotiating insurance claims and helping their clients get the maximum amount possible. If nothing else, hiring an attorney may allow you to stay off the phone and stay with your child.