Vehicle showed up in the early seventeenth century, and back then it meant “a medium through which a drug or medicine is administered”. Can you believe it? It’s from the classical Latin vehiculum, which is vehicle and from veher, to travel. That word is from the Proto Indo European wegh, to move or transport. That also happens to be the origin word of wagon, if by a different means. See, wagon showed up in the late fifteenth century from the Middle Dutch wagen and Proto Germanic wagnaz, which is what comes from wegh-.
Car actually showed up in the fourteenth century, although back then it was just a wheeled vehicle. Because, you know. No engines. It comes from the Anglo French carre and Old North French carre, which is then from the Vulgar Latin carra, related to the classical Latin carrum/carrus, which mean handcart and vehicle, respectively. It can actually be traced all the way back to the Proto Indo European kers-, to run, and is related to words like carousel, carriage, chariot, and charge. Also carpenter. But that’s a story for another day.
Truck didn’t show up until the seventeenth century—there’s another version of it that means barter or exchange that came several centuries earlier, but it’s not related to this one. The vehicle one is thought to be from the classical Latin trochus, an iron hoop, which of course was taken from the Greek trokhos, wheel, and trekhein, to run. So truck was originally a wheel, then it was the thing that the wheels were on.
Van didn’t show up until 1829, although that’s because it’s short for caravan. Caravan is older, having shown up in the late sixteenth century from the Middle French caravane, Old French carvane, and Medieval Latin caravana. The word was picked up during the Crusades from the Arabic qairawan and Persian karwan, a group of desert travelers. And now it’s the soccer mom’s vehicle of choice.
Sedan showed up in the midseventeenth century meaning a covered chair on poles. I think I remember a sedan being a chair in something I’ve read over the years, although I can’t remember what. Anyway, it’s thought to be from the Italian word sede, which means seat, which is from the classical Latin sedere, to sit, and can be traced to the Proto Indo European sed-, also to sit. As to why it became a word for a type of vehicle… because you sit in them?