The only reason I could think of not to reveal would be the fear of collusion. If they know the comparison stores are Alice’s Deli and Bob's Bodega, they could either collude with those owners to raise prices on select items or simply go and look at prices and only offer copies of the most marked up items in those stores (which could be as simple as loading the sandwich with cheap toppings that would be extra at the comparison shops).

The policy says lowest price. Not lowest price of three specific vendors.

So if the three vendors names tried to collude, then they would no longer be the lowest prices.

Any sane policy would rotate data points. How many sandwich shops are there in NYC, hundreds? Thousands even? How often do they shut down, open, change owners...? Burning a couple every year is not an issue.