This is an interesting question and one that is pertinent to American society. As you know there are many places that sell food and the price differences can be very different. So, as a general rule of thumb, people with less money and people who are on food stamps buy their food at discount food vendors. Each state and city will have different types of discount food stores. However, there are some more popular ones.
For instance, one of the great discount stores is Walmart. And many Walmarts have food as well. This would be a great place to buy food at discount prices. There are also places like BJ Warehouse and Costco. These store are able to sell food at a reduced price because they sell in bulk. Though people using foodstamps aren't given enough (unless they have many children) to be able to pay the higher unit price of a bulk package. There are also more regionalized supermarkets that sell at good prices, such as C-Town and Shoprite. All of these stores take foodstamps.
Speaking as someone who at one time sustained a family of four on Basic Food Benefits, we did our shopping at two or three places at a time depending on who had the best prices that week for what we needed, and which coupons I had collected. Costco, sadly, does not accept EBT (or at least they didn't at the time), so that was never an option for us. I would spend several days scouring as many circulars, ads, coupon websites, and apps as possible in order to gather coupons and to find the best deals. I would then divide my shopping list by store, and hit each store in order of furthest from my house to closest. We would use our own cloth bags, because at the time it was still a rather novel concept and most stores would offer a $0.5 discount per bag (which doesn't sound like much, but when you're working two jobs and still need help from food stamps to feed your kids, every penny helps!). For us, the best deals came from Safeway, Fred Meyer, and Target.
Of course those who are food stamps have a tighter budget than most and some times the amount they get does not even cover the cost of food for the entire month. So they will seek stores that have deals like buy 1 get 1 free deals or 10 for 10. In my area it is stores like Publix and Walmart, yes regular stores. In reality there is no special type of store that people with EBT Snap shop at, if anything they shop at stores that everybody shop at.