By the time we have things we call "supermarkets" in the 1950's, both in Europe and the United States, there were already fresh fruits and vegetables available any time. Now, exactly which fruits and vegetables those are would vary from place and place and season to season, but certainly something fresh was available year round. Even older green-grocers decades before then usually managed to have something fresh most of the year.

But, if you're thinking of the usual spread of vegetables available in a Vons (is that a national chain?) or a Tesco's, like tomatoes, cucumbers, lettuce, onions, garlic, potatoes, peppers, maybe a few others, and fruit such as apples, bananas, oranges, grapes, and maybe some stone fruit or tropical fruit, the things that you pretty much expect to find whenever (and my students have NO IDEA when these things are in season, or even that they HAVE seasons) then basically we're talking about the postwar period. There was long-distance shipment of fresh goods before World War II, and even before World War I for some items like oranges, which were shipped from California and Florida to the eastern metropolis. However, these things aren't really fully available year-round and at any large grocery store in Europe and the United States until after the war.

The large-scale infrastructure necessary was likely in place before 1945. Susan Freidberg has described the creation of a "cold chain" of refrigerated spaces like railways cars and warehouses connecting farms and markets in the late 19th and early 20th centuries, though I believe that was typically used more for eggs, dairy, and meat than fruit and vegetables. The other major pieces of infrastructure were railways and steamships, both of which were well developed by the early 20th century. The real changes that came about with supermarkets were in the consumption patterns of families; we could equally think of this as "infrastructure" as well, but of a different sort, perhaps a smaller scale. Consumers had to begin buying things like automobiles and refrigerators, and those are not really common--and certainly not universal--until after the Second World War. Without those items, large supermarkets were impractical, and people shopped mostly at their local markets, many of which remained quite specialized as independent butchers, bakers, and grocers well into the postwar period, particularly in Europe.

See Victoria de Grazia, Irresistible Empire (which covers the spread of supermarkets from the US to Europe, though it is NOT a particularly good book), and Susan Freidberg, "Moral Economies and the Cold Chain" (behind a paywall here, sorry; I'm not sure if she has a monograph on the subject or not. I haven't seen one.).