If a shopping cart from a well known grocery store hits your car, would they be responsible to pay for the damage of the car and a rental if needed?

Not without more than this. In order to recover, you would have to show a breach by the retailer of some duty of care owed to you. If the retailer was careful with the carts, but you drove into one, or another customer was responsible for the damage, then the retailer might not be liable.