I can get it down to £50ish if we don't eat meat, and I reign in the junkfood monster that is my husband. If we order fruit and veg boxes that goes up about £10. Eating meat, I aim for £70-80 including the produce boxes. If I am not careful, it can easily go up to £100. This includes pet and cleaning supplies and most medicines.
We shop mostly at Sainsbury's. I don't mind Tesco, but they don't have all the things we're used to. We'd like to do Waitrose/Ocado, but it's outside our range. We've not been happy with Asda and find their policies with grocery orders crappy. It took a few deliveries to run foul of them, but even the attitude from the person we spoke to on the phone turned us off them. Just as well I guess.
We sometimes try Lidl or Aldi (although there really isn't an Aldi close to us). I don't think we could shop at either exclusively. We just don't eat the types of food there often (ready meals, highly processed stuff). I haven't found their produce all that brilliant. Some things are really great at both though.
We get our veg box through Riverford. When we shop in the actual shop, we sometimes buy discounted stuff, but we must not hit the shops at the right time. We get deliveries because it's a lot easier, and aim for discounted slots or do a big order once or twice a month for free delivery.
ETA: I sound like a total food snob, and I am not. And I love a good ready meal/bag of crisps just as much as the next person, if not more. Maybe that's why I can't live on it or even eat it very often. If I have to cook something from scratch, I eat less often and only when I am hungry. I also aim to have the most nutrition I can pack into the food I eat. When I visit MSE, I marvel at how little they spend on their food, but a lot of them, even the ones that make a lot of their stuff from scratch, eat a lot more sugar, refined flour, etc than we do. We don't eat out often, and instead of doing a weekly meal out, we tend to make something like pizza (or buy it frozen) as a treat. We probably go out for a meal once every month or two.