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According to the Oxford English Dictionary, the first use of the word "van" in print to refer to "A covered vehicle chiefly employed for the conveyance of goods" was in 1829. These were not motor vehicles as we think of vans today, just horse-drawn wagons. See the Wikipedia page for Pantechnicon van for information and a photo of a particular type of English horse-drawn furniture van.
Van was also used to refer to "a closed carriage or truck used on railways for conveying passengers' luggage and the guard of the train, or in goods trains for smaller articles needing protection from the weather".
As Legion600 commented, "van" is indeed short for "caravan." The OED's first citation of "caravan" as "a covered carriage or cart" is from 1674.
An interesting fact to go along with this -- "car" isn't modern, either. The OED cites it with the meaning "a wheeled, usually horse-drawn conveyance; a carriage, cart, or wagon" all the way back to circa 1320.