I was at Costco buying groceries for my dad and the cashier said I could not use his card as my picture was not on the card and I could not otherwise prove authorisation. He said I could be arrested for fraud. I told my sister and asked why she had not authorised me to use the card and she said because no one ever checks and it was a bother. Did she break the law?
If you need to shop for groceries there regularly for your dad, she could use his money to pay for a membership card for you. Or you could just shop for him at a regular grocery store and not have to use a membership card.
Also, just to clarify for the future, your dad's estate is what is left to be settled after he passes away. The person who handles that is his Executor. (It may or not be your sister, depending on what his will says.) The POA gives your sister authorization to handle his matters while he's still alive, as his Agent.