That's weird. I go to local dollar stores all the time to buy some groceries (usually yogurt, cookies, soda and bread) and they all take debit or credit cards.
Of course, most stores down here won't let you pay with one unless you purchase a certain amount of items. The Dollar Tree stores have a $7 minimum if you want to use your credit or debit card to pay. 99 Cent Only stores don't have a limit and I usually pay with my debit card there.
The Dollar General stores I visited allowed me to use my debit card to pay. Weird.