You need to differentiate between the orthographically correct representation and the typographic treatment of text.
For example, a photographer by the name of Robert Smith might decide to write his name in lowercase letters, as "robert smith", both in his signature and in the wordmark representing him in letterheads, on his website and in watermarks in his published photos. If you write an article about that photographer, you would still give his name in the orthographically correct form, as "Robert Smith", with capital letters beginning both the first and last name. You might mention what his workmark looks like ("Robert Smith, who styles his name in lowercase letters,..."), or provide an image of it, but you would certainly not try to represent it in text, except enclosed in quotation marks ("Robert Smith, who styles his name in lowercase letters as 'robert smith', because, as he says, ...").
In English, "when abbreviating a phrase where only the first letter of each word is taken, then all letters should be capitalized" (Wikipedia). Only acronyms "that were originally capitalized (with or without periods) but have since entered the vocabulary as generic word are no longer written with capital letters nor with any periods. Examples are sonar, radar, lidar, laser, snafu, and scuba." (ibd.)
The APA Publication Manual (6th ed., 2009, p. 107) recommends that if you are unsure about wether an acronym is considered an acronym or a word, look it up in Merriam-Webster. If it is an acronym (or other abbreviation), it will be marked as "abbreviation" (e.g. http://www.merriam-webster.com/dictionary/usa). If it is not marked as abbreviation, it is considered a word and not capitalized. Exceptions are the beginning of sentences ("Radar is a technology...") and "spelled" acronyms (such as IQ, noun). If an acronym (or other abbreviation) is not in Merriam-Webster, it is not a word but an abbreviation.
In your example, you must give the name of the organization as "NECOCC", and do not forget to explain the abbreviation the first time you introduce it ("The National Ecological Council of Concerned Citizens (NECOCC) have... Yesterday, NECOCC released a press statement that...").
In any kind of non-formal writing, you are of course free to do however you wish.