At a store without requiring a bank account, you can add cash to your Cash App account only with a Green Dot MoneyPak card. Your Cash App account is linked to the card after you load cash onto it. I found a lot of misleading information when trying to research how to properly do this online, including the claim that you can load money onto your Cash App account at the cash register in a store (which is not true).
To know more:
Where can I load my Cash App card