No car insurance does not go onto your credit report nor does making monthly payments on your car insurance improve your credit score.
Even shopping for car insurance will not harm your credit score even though the car insurance quote will pull your credit report it will only be a soft inquiry that does not affect your credit score.
If you fail to pay for auto insurance then they will just cancel you and will unlikely report it to the credit bureaus.
So it doesn't matter whether you pay for auto insurance monthly or yearly but I just pay my auto insurance payments yearly because I save money by doing so.
I'd be paying that money anyway over the months and it saves time as I do not have to continue making payments for the auto insurance.
Paying the auto insurance yearly saves me time as well as money.