No, you do not have recourse. Stores cannot be held liable for shopping cart damage to cars. They cannot control where the customers put the carts, and they cannot control the weather. This would be considered an "act of God" and therefore they are not liable.
Just because something bad happens at a business (a person or property is injured), it does not mean that the business should be held accountable, our society is too litigious as it is. Your insurance should cover any damage, with a deductible of course.
Last summer somebody hit my car in my apartment parking lot and drove off. I had to get a panel on my door replaced and they had to paint into the front bumper and back door, I was without my car for over a week. Since it happened when I was not there, I was not able to find out who it was. Should I have made a claim with my apartment since the damage happened in their parking lot by presumably one of their tenants? No. They cannot control the actions of other people. Did it suck having to pay a $500 deductible when this was in no way my fault? Absolutely, but that is life sometimes.
In order for a place to be liable, there must be a breach of duty. If Kroger had a corral for the carts and sent people out periodically to bring them back in, then they have met the expected standard of care. They cannot be expected to sit there in the parking lot and personally grab every cart that somebody doesn't feel like putting in the corral as soon as it happens. That is ridiculous. You can certainly try to pursue it, but I highly doubt you have a case. It sounds like it is time to cut your losses and move on.