While a definition relating to organic farming itself is given, definition #3 is likely the origin of the term.
From google:
Organic
1. of, relating to, or derived from living matter.
"organic soils"
synonyms: living, live, animate, biological, biotic
"organic matter"
Chemistry
of, relating to, or denoting compounds containing carbon (other than simple binary compounds and salts) and chiefly or ultimately of biological origin.
(of food or farming methods) produced or involving production without the use of chemical fertilizers, pesticides, or other artificial agents.
synonyms: pesticide-free, additive-free, natural
"organic vegetables"
2. Physiology
of or relating to a bodily organ or organs.
Medicine
(of a disease) affecting the structure of an organ.
3. denoting a relation between elements of something such that they fit together harmoniously as necessary parts of a whole.
"the organic unity of the integral work of art"
synonyms: structured, organized, coherent, integrated, coordinated, ordered, harmonious
"a society is an organic whole"
characterized by continuous or natural development.
"companies expand as much by acquisition as by organic growth"