There are many ways to handle a story in a dark ride. The easiest way is to forego story altogether, as the earliest rides did: just pack in the scares under a suitably generic name like "Zombie Castle." There's an opportunity here to organize those gags in some way, and it's common to sort them based on when one might encounter each one in a haunted house: the front yard, the conservatory, the basement (often a dungeon) or attic, the backyard (often a cemetery). For all its technical sophistication, Disney's Haunted Mansion follows this basic plot.[^1]
[^1]: Over the years, guests have found patterns and relationships that assert a more complex story on the Haunted Mansion, but those patterns weren't really there. Nevertheless, Disney has promoted some of them to canon, bringing the ride's plot up to a slightly higher level of sophistication.
Sometimes this idea is taken half a step forward, replacing "scary" with some other, more specific theme or concept, without going so far as to introduce an actual storyline. Garfield's Nightmare at Kennywood is one such example, intermingling almost a dozen Garfield-related plywood dioramas with giant-sized comic strips ("scary" becomes "Garfield"); "it's a small world" at Disneyland is a rather different example, packing in hundreds of childlike dolls, representing almost every country and culture on Earth ("scary" becomes "children of the world").
If you're looking for something with more of a beginning, middle, and end, you may consider adapting a story your guests already know. The Fantasyland dark rides at Disney parks take this approach exclusively: they pick a Disney animated feature, select about a dozen moments from that film, create a gag for each one, and send a cart whipping along past each one in turn. This works fantastically well if your audience already knows the story you're about to tell them, but it's unlikely that your project fits into this mold. (None of mine do, at least.)
We've gotten this far and still don't really have a precedent for telling a complete story in a dark ride. In general, dark rides usually don’t tell a story. Instead, narratives that employ the medium tend to follow a trend. Marc Davis, via Passport to Dreams, tells us a bit about this trend as it related to Pirates of the Caribbean:
[Walt Disney] didn’t like the idea of telling stories in this medium. It’s not a story telling medium. But it does give you experiences. You experience the idea of pirates. You don’t see a story that starts at the beginning and ends with, “By golly, they got the dirty dog.” It wasn’t that way.
The dark ride has almost always been treated more like documentary than drama. Even when Disney is adapting a well-known story—Peter Pan’s Flight, for example—the plot is whittled down to a few memorable scenes, frozen in time, and overlaid with relatively disconnected, easily-looped lines of dialogue (“Look out, Peter!” “Here we go!”). There is a beginning, middle, and end, but in the broadest strokes possible.
Marc Davis says it’s not a storytelling medium, and while he has the résumé to speak with authority here, I think there's room to expand. I understand the view: if theater evolved from conversation, then the dark ride evolved from nature. That is, an “organic,” spontaneously-occurring play is just a few people interacting with each other, and an organic dark ride is a place that would exist regardless of whether you were moving through it on a track. An argument vs. a garden path. An argument rises and falls and eventually resolves, while a garden path merely twists and eventually ends.
In Pirates, the story is not so much one of interpersonal conflict as conflict against one's environment. The pirates succeed in overpowering their targets without much difficulty at all, and proceed to do as they please for as long as they like. It's a satisfying tableau, in a sort of hedonistic way, but I think its lasting power is in the end—and then, again, in the beginning. The last time we see the pirates, they're on the verge of incapacitation, consumed by their own success, while the wrecked town burns down around them. They carelessly fire bullets in victory from behind droopy eyelids, ignorant or simply indifferent to the explosives surrounding them. These pirates succeed but they don't win, and as we float back up the waterfall, we're encouraged to remember how this all started: forgotten skeletons perched on stacks of treasure. There is no glory here. Dead men tell no tales.
This "idea of piracy" and the scattered narrative that Walt Disney called a "cocktail party" has certainly proven its lasting merit. Inundating your guests with a pile of little stories lets them pick and choose the plot lines they deem important, and in so doing they get everything they want. Furthermore, Pirates seems to place its conflict in a meta-narrative, leaving even the final act of the story to play out only in guests' minds as they wander through the gift shop at the end of the ride. It's a fine model for a dark ride—perhaps the best model—but it can't be the only one.
In later decades, another trope took hold that was nearly the opposite of what Pirates accomplished. In these "modern" rides, something has gone wrong, and the hero needs your help to fix it—yes, you, right there. Inevitably, you would save the day, simply by sitting down and keeping your hands inside the vehicle.
This newer trend seemed to be pure conflict and exposition. Placemaking was secondary to the big, urgent problem at hand, and all we're here to do is get in, fix whatever is wrong, and get out. This style of ride tends to be short, and when it's not short it's only due to a series of failed attempts at solving the problem. There's exactly one narrative, it's fed to every guest, tied up with a neat little bow, and tossed out at the end. You'll probably get the whole thing on your first ride, and every ride after that will be more or less the same. If you find a ride in this style that you love, you might try to keep riding it until you know the story so well that it fades into the background, and you can just focus on the dinosaurs and the thrill of the vehicle whipping around, but even in the best case scenario, you're acting against the ride's intent.
Still, even though it’s a stretch from the natural state of the medium, I’m confident that a coherent, specific story can be told like this. A dark ride can be like a safari but it can also be something like a picture book and a radio play combined into one. Very few works in the medium have tried this, but it doesn’t mean that it can't work.
We might consider looking to children's books for inspiration here. Sam & Dave Dig a Hole by Mac Barnett, for example, has a beginning, middle, and end, but it's very light on exposition. There are probably just dozens of words in that story, and they revel in repetition and simple language. However, there's a parallel narrative in there, told entirely in subtext—so subtextual, in fact, that it's only in the illustrations. There's next to nothing by way of character development, but with no particular personalities with which to paint Sam and Dave, we become them. In Understanding Comics, Scott McCloud says that TKTKTK detail in drawing faces
We can look at one of EPCOT's landmark dark rides, Spaceship Earth, for a hint at how we might pull this off. This is a story about humans, as told through major breakthroughs in technology. We start in a cave, discovering fire and making marks on the rock walls; soon, papyrus, and the printing press, and personal computers. It's a story about progress, with no single protagonist, and the only trait that all the characters have in common is that they're human, like us. Unlike Pirates of the Caribbean, we are always focused on one moment, and through composition we are usually coaxed into paying attention to just one particular person at a time. EPCOT tends to lean toward the pedagogical, opting for information over drama, and so the only possible antagonist here is time itself, stopping us from discovering the future any more quickly than we already have. But what would this ride look like with a villain?