It is possible to say "eat the apple" and be perfectly grammatical in English but the meaning would be that a single apple was being shared. I don't suppose that is how this sentence would be understood in Vietnamese. The question that occurs to an English speaker is whether the Vietnamese can have both meanings.
What is correct depends on what the Vietnamese actually means. The various English sentences are acceptable as English but they mean different things. "The children eat apple" is less likely to be said in English than other possibilities. The question is what does the Vietnamese mean or can it mean all of the various English meanings.
I will argue that the correct translation of this sentence is actually "The children eat apples." There is nothing that specifies how many or which apples the children eat, so it is more correct to just say they eat apples (in general).
The usage of 'the' points out a specific apple. Since Vietnamese has no article 'the', the best translation of 'the children eat the apple' in Vietnamese that I can come up with would be 'the children eat the apple over there' or 'the children eat that apple': những đứa trẻ ăn táo đó.
"the children eat apple" should be counted as correct. It is not specified as being either "the" or "an" apple. Why is it OK to say "the men eat papaya" or "the girl eats bread" but not "the children eat apple"? Just that it is an uncommon way of saying it in English doesn't make it incorrect
There is a grammar problem in the English because there would have to be an article (a, an, the) with a singular English concrete noun. There is, however, no problem in English grammar with saying, "The children eat an apple, " nor is there a problem with the Vietnamese grammar. The problem is one of comprehension of a Vietnamese noun as plural when there is NO AUDIBLE OR VISIBLE INDICATION OF PLURALITY IN THE WORD OR THE SENTENCE. The requirement to interpret the sentence by sense rather than form is new and confusing to English speaking learners.
The problem is that "The children eat apple," in English is different from "The children eat an apple," or "The children eat apples." It is posible that someone might say "The children eat apple" but it would make apple an uncountable noun like 'steak' (which native speakers do not do) or it would mean that apple was the flavor of choice from among multiple available flavors. (Pie, juice, etc.)
Maybe in New Zealand that might be the case, but here in Maryland that's "The children eat an/the apple" with an article being de rigeur according to how the speaker regards the apple. "The children eat apple" means "apple" as an option among choices, e.g., apple pie as opposed to cherry pie. In that case, "apple" is actually attributive to some omitted noun.