Re: Fender Bender in Rental, No Insurance :(
Read your rental contract and it will explain to you exactly how their insurance works. As far as I know, and unless you pay for their insurance, they as well as the other party would have a valid civil claim in case of an accident.
In other words, and if the other driver was to go through their insurance for the repairs, there is a good possibility that Avis' insurance will come after you for the amount of the claim that they paid to the other driver.
If in fact they failed to enforce their policy of "no insurance/no rental unless you buy ours", and even if that means they should not have rented to you, I highly doubt that it will your saving grace... You can still be held liable for damages to the other vehicle.
Also, keep in mind that your reporting the accident to them, might suggest that they will report it to the DMV. That will put you in a double bind... One, you will be held liable for the damages to their vehicle as well as the other vehicle and TWO, and since the DMV had no proof of insurance information covering you while you drive, your license can end up getting suspended for being in an accident while driving without insurance.
With all that being said, I am by no means suggesting that you should not report it to Avis, nor am I advising you not to report it to the DMV (if the damages exceed $750). I am simply outlining your options along with the possible consequences stemming from each. The choice as to which way you choose to proceed, is ultimately yours!
I am right 97% of the time... Who cares about the other 4%!