This has been tried a lot in the 20th century (starting in the 19th century), usually emerging from the labour movement. I think in food retail and energy it has mostly failed or simply stopped making difference. The profit margins of food retailers are small to begin with and easily lost by being a bit less efficient.
But Europe and the UK in particular has been using a similar model pretty successfully to keep drug prices low. Some argue that costs were merely shifted over to the U.S. I think this is only partially true.
(Well, the actual question is DO they have a monopoly, to which the answer is mostly no)
Target has a NET 2.6% profit margin, Walmart has 1.9%, which are on the low side of what you'd expect for a sustainable commodity business. Tesco appears to have a GROSS (not net: this is after cost of goods but before salaries, etc.) margin of 6.5% and Sainsbury's has 7.6%. [1] Obviously their net profit margin is going to be less. This is hardly a company with a huge pricing power dynamic. If you want to see that, look at Apple (24.5% NET margin), Coca-Cola (22.2% NET margin), Visa (50.3% NET margin). (Note however that Coca-Cola's customers are its bottlers and Visa's customers are merchants so their pricing power might be invisible to end consumers)
Gross is 23.5%- it makes you wonder where all that money is going.. oh yeah, $9.85 B of stock buybacks.
Why can they do this? One reason is monopoly power- monopolies must be broken up.
But what else can be done (besides central bank driven demand destruction)? So instead of price controls, a more palatable solution is collective buying. This gives buyers more clout, allows them to bargain for lower prices.
https://en.wikipedia.org/wiki/Buyers_club
So we need more large non-profit buyers clubs, things like that. In the US, allow medicare to bargain with drug companies- it's insane that they can not.
An example where this works: my parents live in a gated community. They pay very little for the cable + internet + cell phone because the entire community has a deal with the provider.
Large retailers have this exact role: they certainly demand and get lower prices from their suppliers. We need a non-profit version of this, why should Walmart owners only benefit from their buying power?