Thats a shame because if it accepted 'the shopping' or 'the shop' it would feel like a better translation to me. Most English speakers I've met would understand that if someone in the household is doing 'the shop' they're getting in groceries probably for a few days or a week
"The man/husband does the shopping" is also a valid expression, and is more common than Duo's answer. I have never said, or heard anybody say, "I'm going to buy/buy in", but rather, "I'm going shopping".
I don't know the translation in english. I would say "go shopping" Købe ind is not specifically used for food. It is the generic everyday shopping :
When you go to the supermarket on saturday to buy various things for the week, socks, toothpaste, food, shampoo, beer, zucchini, light bulbs etc...