What law mandates supermarkets to scan their own cards?

I was told that legally, supermarkets that have some sort of 'rewards program' in which customers may obtain a discount card with which they may receive discounts on products are required to, when asked, swipe their own card to provide you with the same discounts, regardless of whether or not you are a 'member'. What is the actual law that dictates this?

There is no law, it is store policy.

A lot of the time these rewards programs are for marketing purposes and are designed to collect information, opting into the program give you discount as a trade off for collecting information.

The store is under no obligation to provide rewards to people who have not registered for this program.

I've never heard of such a law; if one exists, it's probably specific to your state.