How/why was the word "organic" chosen to represent natural foods or foods without chemicals?

Solution 1:

Organic as a chemical description

The results of a Google Books search for the phrases "organic foods," "organic diet," "organic farming," "organic gardening," and "organic vegetables" in works published between 1800 and 1960 corroborate the poster's impression that notions of organic and inorganic chemical composition preceded usage of organic in the sense of healthful, naturally occurring, and/or unadulterated.

Matches for "organic foods" in a Google Books search go back as far as 1853. From “Agricultural Chemistry,” in Encyclopedia Britannica, eighth edition (1853):

Carbonic acid, ammonia, and water, are the great organic foods of plants. But while the plant has afforded to it an inexhaustible supply of the last, the quantities of the two former, both in the atmosphere and the soil, which are available as food, are limited, and insufficient to sustain its life for a prolonged period.

An explanation of "organic foods" appears in An M.D., “The Theory of Food,” in The Englishman’s Magazine of Literature, Religion, Science, and Art (September 1865):

Every one knows, I suppose, by this time, the great classification of organic foods which was propounded by Liebig, and which has since been generally accepted. He arranged them under two heads — the heat-creating, and the tissue-forming foods. The latter, as they one and all contain nitrogen, though in varying proportion, were called nitrogenous, or nutritive ; the former, calorifacient, being for the most part destitute of nitrogen, and containing chiefly carbon, hydrogen, and oxygen.

The distinction between organic and inorganic is more fully addressed in W. H. Broadbent, “An Attempt to Apply Chemical Principles in Explanation of the Action of Poisons,” in Proceedings of the Royal Society of London (June 18, 1868):

The operations taking place in the animal organism may be divided into two great classes :—(a) for maintenance of structure, (b) for evolution of force. ...

The two great classes of food, organic and mineral, are in close relation with these. The organic foods build up the tissues, but ultimately undergo oxidation and yield force. The inorganic foods take a subordinate part in the composition of the textures ; they do not yield force by oxidation, but they influence the nutritive processes.

The phrase "organic diet" likewise appears in some relatively early serach results. From Dr. Elam, “Popular Physiology,” in The Eclectic Review (January 1861):

The vegetable cell attracts to itself certain portions of water, carbonic acid, and ammonia, whence it extracts oxygen, hydrogen, carbon, and azote, and from these it constructs an albuminous pabulum which serves it for nutrition. The animal cell appears incapable of extracting its nourishment from the inorganic world, and requires absolutely that an organic diet be previously prepared for it. This it attracts to itself, and operates upon until it be fit for its nutriment.

And from John Hobson, The Social Problem: Life and Work (1902):

So, while each rational man, with what small aid “science” can afford, must build up a complex organic diet for himself adapted to his individual needs, and changing as those change, he must similarly work out a complex standard of uses of energy. If he is wise, he will shun exactitude and leave "a broad margin to life," a large proportion of his fund of energy and time unallotted to specific purposes ; the rest he will distribute so as to get the widest variety of exercise consistent with such specialization as is demanded by society, or for some private satisfaction.

In all of these instances, organic appears in contrast to the alternative inorganic; and in most of them, it is itself subdivided into nitrogenous and non-nitrogenous forms. But the notions of "natural" and "unadulterated" do not enter into the analysis. Not coincidentally, there are no relevant (and confirmable) matches for the search terms "organic farming," "organic gardening," and "organic vegetables" from 1800 through most of the 1930s.

Organic as 'natural,' 'healthful,' or 'unadulterated'

A 1945 match for "organic gardening" describes why people adopted the word organic to refer to farming and gardening techniques that avoided using chemical additives such as pesticides and artificial fertilizers. From The Fruit Orchard (1945), one in a series of books on organic gardening issued by the same publisher [combined snippets]:

Organic Gardening Series

This series contains the most valuable reprints from back numbers together with much original material first printed and specially prepared. Organic Gardening is the science and art of feeding plants with organic materials, to produce healthy plants, so that we ourselves can obtain healthy food.

The crucial point here is that the people in this movement saw themselves as supplying organic foods (in the nineteenth-century sense of that phrase) rather than inorganic ones to the plants that they were growing. As a result, an organically farmed vegetable is organic not just in the sense that it contains carbon, hydrogen, oxygen, and nitrogen, but in the sense that it doesn't contain artificially abundant quantities of various inorganic elements, because of the way it was grown.

To like effect is an earlier item from Feed & Farm Supplies (1941) [combined snippets]:

In many parts of the world, where settled agriculture has only existed for a few decades, experience is necessarily lacking and the unco-ordinated application of modern agricultural technique has had disastrous consequences for the soil. People are more and more turning to "organic farming" looking to natural plant covers and organic manures to preserve the soil fertility which human inexperience only destroys. In this country, also, we tend to look back rather wistfully to those days when organic manures and a few recognized crop rotations sufficed to keep the land in good heart for agriculture, although at a level of productivity that would be inadequate to meet our present and future requirements.

The movement took off during the 1940s—and by the 1950s, the new understanding of organic was well established. From The Organic Farmer: Farming Without Chemicals, volume 3 (1951) [combined snippets]:

Some days we have a very busy driveway, with all sorts of cars and trucks coming and going. One day when I was trying to freeze spinach we seemed to have a stream of cars. The first was an organic gardener from a nearby town who wanted spinach for her freezer. After she went a truck came up filled with boys from the city who came after rough lumber for the church camp. Later in the day three more cars came, each for spinach. As usual I always stop to talk and take visitors out to the gardens to show them what we are doing in the organic line. I show them the peas coming up in thick straw mulch, a patch of potatoes mulched with straw and another with sawdust. Then I always have to show them the coolness of the soil under the mulch and the countless worms and tunnels. Then we talk of the wonderful life in the soil, how marvelous the vegetables taste, and how much our health has improved since eating organic vegetables. One visitor told us she had had cancer and had been cured. The first thing her doctor told her was to throw out her aluminum ware or his treatment would have no effect. This was about ten years ago and she is still spry. Her remark about aluminum made me throw out all my aluminum ware.

Solution 2:

The word organic means "relating to or derived from living matter" (cf. organism). Although I don't know how long the term has been used to describe food, the supermarket usage of organic clearly comes from this definition.

It was actually the chemists who decided to change to a specialized carbon-based definition, after they learned in the 19th century that certain compounds were not unique to living organisms. According to Wikipedia:

[The term organic compound] is historical, dating to the 1st century. For many centuries, Western alchemists believed in vitalism. This is the theory that certain compounds could be synthesized only from their classical elements—earth, water, air, and fire—by the action of a "life-force" that only organisms possessed. Vitalism taught that these "organic" compounds were fundamentally different from the "inorganic" compounds that could be obtained from the elements by chemical manipulation.

Vitalism survived for a while even after the rise of modern atomic theory and the replacement of the Aristotelian elements by those we know today. It first came under question in 1824, when Friedrich Wöhler synthesized oxalic acid, a compound known to occur only in living organisms, from cyanogen. A more decisive experiment was Wöhler's 1828 synthesis of urea from the inorganic salts potassium cyanate and ammonium sulfate. Urea had long been considered an "organic" compound, as it was known to occur only in the urine of living organisms. Wöhler's experiments were followed by many others, where increasingly complex "organic" substances were produced from "inorganic" ones without the involvement of any living organism.

Even though vitalism has been discredited, scientific nomenclature retains the distinction between organic and inorganic compounds. The modern meaning of organic compound is any compound that contains a significant amount of carbon—even though many of the organic compounds known today have no connection to any substance found in living organisms.