Unless your vehicle was involved, then you would not file a claim with your car insurance company for the damage the car sustained due to the shopping cart getting out of your control.
Depending on your renters/homeowners policy, this is where you could try to file this claim or under an umbrella policy.
When buying a renters policy you want to make sure your policy covers damages caused away from your home. If it does then you should file it under your renters policy, as it appears you are liable and your renters policy may cover you.
Most renters policies have some personal liability protection, many times this situation would not fall under it though either. This is because most renters policies provide limited financial protection against liability claims and lawsuits brought by others for accidental bodily injury or damage to their property, but many clauses require it to occur in the covered area (renters).
Contact your renters insurance agent to find out for certain if a claim can be placed against this policy or not for the damage the other party's car received by the cart. If it does not cover this the other party should be able to place a claim for this against their collision coverage if they have such a policy on their vehicle.
You can obtain renters or homeowners insurance and umbrella insurance with our partner Safeco by following this links or calling them directly at 1-800-936-0356.