Auto accident question

Yes, you can seek compensation: failing to have insurance exposes you to tickets, fines, and possible liability (e.g. if you'd been at fault) but does not mean that an at-fault driver can damage your car or injure you with impunity: she still has to pay for the injuries, medical bills, card damage, etc. she caused. 
However, you don't "go after" her insurer: her insurer has no obligation to you--it is her insurer, not yours, after all. Her insurer's obligation is to her: to defend her in court and to pay on her behalf if she is ordered to pay money. Sometimes they voluntarily offer to pay, if the believe that is more cost effective than fighting on their client behalf, incurring legal bills, then possibly losing and having to pay for her anyway, but is voluntary on the insurer's part to pay unless their insured is sued and loses.
So you can look to submit a claim to the other driver's insurer, and they may choose to offer you a settlement. But if they do not, you'd have to sue the driver, not the insurer, and in court prove her fault (which you should be able to do if she rear-ended you) and also that the injuries you claim are the result of the accident and the extent of and prognosis for the injuries. 
To sue for injuries--not just the medical bills, but for the injuries  themselves; i.e. for "pain and suffering"--you'd need to hire a doctor who examined or treated you to testify in court, and that can be expenivse: such doctors can charge thousands of dollars for participating in trials. So unless the injuries were severe (which you state they were not), it might be worth suing for injuries--only for car damage and bills.