Right, so down the road from where I live in the NE (of the US), there is a fruit-and-vegetable market. Not a dealer, for pity's sake. :) And yes, fruit and vegetables are also called produce, tonic accent on the pro.
Nowadays, there are smaller super markets that have begun to re-specialize in high quality, locally or regionally produced fruit and vegetables, but even so, these establishments typically sell other things as well. Essentially though, the 19th century seller of only fruit and vegetables i.e. a greengrocer as found in a town along a main street went by the wayside or stayed there. Please read on.
Roadside fruit-and-vegetable stands are everywhere on secondary roads that go by farms, and this is the case historically as well. And some farms, do have large facilities selling fruit and vegetables and these are called farmers' markets. Typically, a wooden building filled with stands of fruits and vegetables. Also, cities and towns are holding farmers' markets on the weekends or in the summer. My town has one in a park's parking lot that runs from June through early September. Besides fruit and vegetables, there are also locally produced cheeses, jams and some pottery.
Here is a well-done historical document on this topic from the USDA: farmers' markets The first part of the document has great pictures of roadside farm stands that, in fact, look pretty much like the roadside stands of today.
The document also says this about farmers' markets:
Throughout the United States, the number of farmers markets continues
to rise, their popularity spurred by consumers’ growing demand for
locally produced food. More than 4,900 farmers markets operate
nationwide today—a jump from fewer than 1,800 only 15 years ago. In
addition to providing easier access to fresh food, studies show
establishing a farmers market can revitalize a neighborhood, enhance
social interaction, improve the local economy, and provide a
supplemental source of farm income for many growers.
in turn, that document quotes this book:
Direct Farm Marketing as a Rural Development Tool (1997). Gale, Fred Rural Development Perspectives, 12(2), pp. 19-25
“Farmers’ markets are, of course, the oldest and most common type of
direct selling. A 1993 directory of farmers’ markets published by the
USDA listed 1,755 operating markets. The total number of farmers’
markets may actually be much larger, since this was not an exhaustive
list and probably excluded many smaller markets. Marketing specialists
at USDA and land-grant colleges believe that the number of farmers’
markets is growing, although there are no historical statistics for
comparison. Markets vary widely. Some are year-round, others are
seasonal; some are held in permanent indoor facilities, others are
held in parking lots.” p. 20
In the US, small-scale butcher shops and greengrocers were replaced by large supermarkets at some point around the mid-twentieth century. So, it is not that the term is necessarily British, it's that main street shops called greengrocers went the way of dry goods' stores and local emporia. large supermarkets