I wouldn't turn it in... but I would leave my name and contact information with the store, so that if someone came looking for it, asking about lost money, they could contact me and identify it. I don't trust the person behind the counter to ensure that it goes in their safe, and if it does... and no one claims it, I don't trust them to contact me to receive it as a finder.
I've found cash in stores, and debit cards in parking lots, and each time, I go to the customer service counter and leave my information and general info that I have found money/card.
Once, I found someone's wallet in the top section of a grocery cart in the parking lot. You know the kind... left in the parking space? So I went to push it inside with me and use it and there was a wallet in the child seat area. I opened it up and it appeared to be an elderly woman's wallet, with cash and IDs and credit cards and SS stuff. I took it straight to the customer service counter, told them where I'd found it and said there was a bank card in the wallet so they could call the bank and ask the bank to contact the woman. I did not find any contact info (address, yes, but not a phone number) to contact her directly myself.
Poor woman probably got home and didn't even know she'd lost her wallet until the next time she went somewhere. :(
--
There was a time when I would have turned in a credit/debit card. But not anymore. My husband has left his places before, and once they stick it in the safe it can wipe the magnetic strip and it is no good. And if it doesn't make it to the safe, there are a zillion places a card can be swiped and used without ID. My regular grocery only makes you sign if your credit card is used for a purchase over $25, for Target it's $50, and I don't know about Walmart. And lots of gas stations would work. You'd likely only need the zip code, and being in a small area, that would be easy to guess correctly.