The governor of New York state, Andrew Cuomo, announced today that he will issue an "executive order" (a legally-binding instruction by a senior elected official in a U.S. jurisdiction based on either constitutional or statutory authority) which empowers store owners to refuse entry to anyone not wearing a face covering. "We are giving store owners the right to say, 'If you're not wearing a mask, you can't come in,'" he said at his daily press conference, which has become a popular nationally-televised news event.
I'm not familiar with New York law, and can't assess what the governor's independent authority is in matters such as this, but since the executive order will leave the decision whether to require a mask to individual business owners I suspect a lawsuit challenging its validity is unlikely.
Where I live, the pandemic restrictions have not yet been lifted. I went to a supermarket yesterday for the first time in two weeks―I am in Maryland, in the same county as Alan Goldhammer―and the chain supermarket outlet we typically frequent had an employee at the entrance verifying that everyone entering had a face covering, as well as several others, stationed near the checkout counters, who apparently were doing the same as well as making sure that those in the checkout queue were properly spaced.