Reward programs can link purchases to you through 2 different means:
- if you give your name and phone number to the smartphone application
- if the supermarket links the purchase to the actual bill sent to the bank account
You cannot technically avoid the second one but I know that most stores don't do that because their systems are separated: one system for the inventory, one system for sending transactions to the banks. I'm telling this because, while they can link your debit/credit card to your purchase, it never happens because it's a major PITA for them and it's a manual process. But it can happen...