I think...I think probably, you could just give them the routing number from your account. Every account has to have one, since that's how they get the money off of your card. If you don't know it, you can call your bank and ask them for it. You'll have to verify your identity before they give it to you, but then you'll know what it is.
As for the inability of the store to take credit/debit? That's not the store's choice. Trust me, I'd take your payment with credit or debit in a heartbeat. The only thing I can say is that if you know you're coming in to make a payment, stop by the bank first and withdraw some money. It can't hurt to have a bit of cash on your person, and will make the experience much less painless.
The reason that the stores are unable to accept your credit/debit card is that it charges us a fee for every card we accept. Apparently the corporate office doesn't think it's a good idea to get charged money when you're giving us money back. Which honestly? I understand. I know that it costs .35 a card that's slid through the pin pad at my register. And that's .35 to the company. Now imagine if every customer with a Kohl's charge came through and wanted to pay on their account with a card. We would be losing money faster than we're making it, as we don't count payments as money made...