Can I file an auto accident claim without a lawyer?

If your claim is for car damage and rental, you should be alright.  If you are making a personal injury claim, you should consult with a lawyer.  Insurance companies tend to scrutinize personal injury claims more readily which could catch you off guard.

The short answer is "Yes", you can file an auto accident claim without a lawyer.  The problem is, by doing so (if you suffered bodily injury), there is a strong likelihood that you could harm your case, unless you have been trained and are experienced in how to handle accident claims.  Claims adjusters know what to look for in order to minimize any settlement/pay out.  Far too often, people attempt to handle the claim on their own.  But when the insurance company offers them a very small settlement, they then attempt to retain an attorney.  Many attorneys, at that point, are reluctant to step into the case because of the damage already done by the claimant.  Accordingly, at a minimum, you should consult an experienced attorney to discuss your claim.  Then, if you decide to represent yourself, for better or worse, you would have at least had your questions answered by a legal professional.