No. Whoever negligently didn't secure the cart is responsible.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Runaway cart cases can rely largely on whether a store had a procedure for clearing the lot and whether they followed that procedure. That's why you see kids out in the parking lot hauling carts in when you go to the grocery store or XYZ-Mart. When people who have had damage due to a runaway cart bring a claim, the store defends by saying they have a procedure, they followed the procedure that day [they come up with some sign-off sheet] and so it was out of their control. So, I agree with attorney Koslyn's answer.
As a practcal matter, in a cool calm rational manner you might want to bring this up with customer service. They may or may not want to help you. Beyond that, you could think about going to small claims court.
This answer is provided for informational purposes only. Actual legal advice can only be provided in an office consultation by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies.
Sign up to receive a 3-part series of useful information and advice about personal injury law.