If someone accepts a version of a story completely, without questioning any part of it, they ..... what?
I'm looking for a word or a phrase or a more elegant, eloquent way of saying it.
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It's not particularly "elegant" but the term I think of is "bought it" ("buy it").
3 [transitive] informal to believe something that someone tells you, especially when it is not likely to be true:
Note the "informal" on the definition.
I told him my name was "The Great and Powerful Oz" and he bought it.
Means that the subject believes that my name is "The Great and Powerful Oz", despite the fact that it's a ridiculous thing to say.
I think you want the phrase wholesale acceptance.
For example, one might say or write: "The shooter's wholesale acceptance of white supremacy may have made him an easy recruit for neo-Nazi groups like Stormfront."
Cambridge Dictionaries Online
involving everyone or everything; complete
What the system needs is wholesale reform.
If I read your question right, you're looking for an elegant verb, and hopefully you'll allow me to add an adverb. Here are a few ideas.
They accepted it without question.
They accepted it naively. (If you want to be fancy, naïvely, with two dots over the i.)
If you want to stay away from accept for some reason:
They admitted it unreservedly / without hesitation.
They granted it unquestioningly.
They received it ingenuously. (Ingenuous means innocent and unsuspecting.)
If you need just a verb, with no adverb,
They swallowed it.
This is elegant by virtue of being simple. It is formal if you omit the hook, line and sinker part. The only possible downside I can see is that it's not one of those fancy words that one might want to use to try to impress.
To indicate that the story was believed in its entirety, you could write that they accepted the story without reservation.
If your emphasis is on the failure of the audience to question the story and their uncritical acceptance of it, consider discussing the credulity of the listeners (the individual listeners are credulous).
Credulity is a state of willingness to believe in one or many people or things in the absence of reasonable proof or knowledge.
Credulity is not simply a belief in something that may be false. The subject of the belief may even be correct, but a credulous person will believe it without good evidence. — Wikipedia
I would say gullible, but there are some great synonyms on thesaurus.com:
being a sucker, believing, biting, credulous, easily taken in, easy mark, falling hook line and sinker, foolish, green*, innocent, kidding oneself, mark*, silly, simple, sucker, susceptible, swallowing whole, taken in, taking the bait, trustful, tumbling for, unskeptical, unsophisticated, unsuspecting, wide-eyed
My favorite is credulous
I recently used it this way: John 8:44 ("You are from your father the devil, and you choose to do your father's desires") has been swallowed wholesale by so many that today many dictionaries define "Pharisee" as "hypocrite." And, if not "hypocrite," many intend it as a slur meaning "charlatan" or "liar."