Some places, like Ralph's and CVS Pharmacy, will let you have the discount card without filling anything out. While very ill, I just took the card and used it without giving them my name, address, etc. During that time, I periodically threw the card out and got a new one with a different number. I did this for germ control because I am on the street to get well. But I also was just too ill and overwhelmed to cope with the paperwork. Still, every penny mattered so I needed whatever discounts I could get.
At some point, I began filling out the information online in order to add coupons to my Ralph's card and get access to additional savings. I get really sick from papers, so I mostly do not keep or carry paper coupons. Digital coupons added to my discount card helped me save money and stretch my limited budget without getting ill or carrying more crap.
I recently applied for Target's Red Card. They have two. One is a debit card linked to your bank account. The other is a credit card. It gives 5% off all purchases made with the card. So not on cash or foodstamp purchases. I applied for the debit card. There is no credit check but I needed a phone number, street address, and bank account to get it. It was similar to filling out a credit card application. I am in the process of rebuilding my life so I can get off the street. I applied in part to get the discount, in part to further my goal of re-establishing my financial identity. I was approved. Yay!