Is it impermissible to purchase items from a store that relies on discounts offered by non-profit organizations to low-income individuals, using a discount card belonging to another person?

It is permissible to benefit from your friend's card if the card's terms allow it, and you can inquire with the issuing authority. However, if the terms stipulate that only the cardholder may benefit from it — which is the apparent case — then it is not permissible for you to benefit from it. In such a situation, she must seek absolution from the responsible authority and pay the price difference, if necessary.