If you use "and", then you're grouping together additives, preservatives, and anything that has been synthesized. Is the commission restricting the use of the word "natural" only if a product has ALL of these elements? Could you call a product natural if it has additives and preservatives, but not other synthetic ingredients?

Since we are treating each item in the list as an individual way to restrict the word "natural", then it makes sense to use "or".

amirhakimi wrote:The commission has directed advertisers to restrict the use of the word "natural" to foods that do not contain color or flavor additives, chemical preservatives, or nothing that has been synthesized.

(A) or nothing that has been: "nothing" makes double negatives; Eliminate.
(B) or that has been: destroys the parallel item list; here parallelism is something like: foods that do not contain.... or foods that has been... which makes no sense. moreover, notice that we have no and/or between the first two items: "... additives, chemical preservatives". Eliminate.
(C) and nothing that is: using "and" distorts the intended meaning. Eliminate.
(D) or anything that has been: CORRECT: parallel list is OK we have three noun phrases and the structure conveys the meaning clearly.
(E) and anything: Meaning issue as choice C. Eliminate.