If you noticed the pony and trap recurring and thought, "gee, anything that shows up so often just has to be a symbol," well, you'd be right. For Arthur, the pony and trap are a flashback to the past. He arrives expecting a car, but:
No car appeared. Instead there drew up outside the Gifford Arms a rather worn and shabby pony and trap. (5.1)
For Arthur, this goofy mode of transportation represents an old-school England in which cars and trains are not relevant and everyone travels in carriages and on ponies. It lets him think that the villagers are just a bunch of hicks who probably go muddin' and cow-tipping—and he might be right about that. But the pony and trap also comes to represent a lot more.
Specifically, it represents being stuck in the past. When he comes to Eel Marsh House, he begins hearing the sounds of the pony and trap on the causeway, but it's not a cheerful trotting sound:
It had somehow lost the causeway path and had fallen into the marshes and was being dragged under by the quicksand and the pull of the incoming tide. (6.6)
He hears this again and again before realizing that it's not real; it's a memory that repeats itself. The pony and trap represents the event that Jennet, the woman in black, cannot get over—the death of her child.
We have a name for this in the 20th and 21st centuries, although the 19th century didn't: trauma. What Arthur hears is a traumatic event repeating itself over and over, because Jennet can't figure out how to process it. She's compelled to relieve the event, almost like a war veteran who can't adjust to civilian life again. Will Arthur be doomed to the same fate? Or will telling this story act like a support group, letting him move on from his traumatic past?
One more thing. The pony and trap also comes up again in a very unfortunate way—it is the contraption that kills Arthur's wife and child. In this circumstance, the pony and trap still represents the pull of the past, because even with all the modern modes of transportation, Stella and Joseph still want to ride the pony and trap as a kind of old-school carnival ride—like wanting to ride on a steam engine train, or go for a drive in a Model T. But the ride turns out to be the exact opposite of charmingly nostalgic:
They had almost come to a halt when they passed the tree beside which the woman in black was still standing and, as they did so, she moved quickly, her skirts rustling as if to step into the pony's path. (12.25)
In a truly horrifying instance of the past coming back to haunt Arthur, Stella and Joseph die in a pony and trap accident—almost, but not quite, repeating history. Unlike Jennet, Arthur will be able to move on from this awful accident without bringing dozens of innocent children down with him.