There are two very useful timelines at the Cold Spot Paranormal Research website. The first timeline looks at ghost hunting in the 1800s, and the second continues this into the 1900s. Michelle McKay, who compiled these histories, was very smart to include some useful context. For instance, she marks key historical periods, such as the Victorian era, and notes important dates in related movements, such as Spiritualism.
McKay and I have both researched when the term “ghost-hunter” first appeared. To date, the earliest sources I’ve found came out in January of 1804. That month, two magazines covered the Hammersmith Ghost, which involves a man being mistaken for the ghost as he walked along “Black Lion Lane, where the ghost-hunters were lying in wait,” to quote those articles. The mistake proved to be a fatal one, and it’s discussed on this page: “Francis Smith and the Tragedy of the Black Lion Lane ‘Ghost Hunters.’“
Prior to this discovery, McKay at set the mark at 1809, when a novel titled Osrick; or, Modern Horrors used the term. Before that, I had traced “ghost hunter” to the 1830s, which appears to be when the term became more popular. In 1833, Michael Banim made quite a splash with a novel titled The Ghost-Hunter and His Family. I’ve come across many reviews of this work, suggesting it was widely read in its day, in turn, making the term “ghost-hunter” popular, too.
A better remembered ghost story from the same decade is Richard Harris Barham’s “The Spectre of Tappington,” which debuted in an 1837 issue of Bentley’s Miscellany. This comical story about a ghost that steals its victim’s pants uses “ghost-hunter” twice in reference to Tom Ingoldsby, who probes the alleged haunting. His initial late-night vigil is a bust, and it’s there that Barham first uses “ghost-hunter.” Tom’s next investigation goes better. Barham writes:
The watch of the previous night had been unsuccessful, probably because it was undisguised. To-night he would ‘ensconce himself’ . . . in a small closet which opened from one corner of the room, and, by leaving the door ajar, would give to its occupant a view of all that might pass in the apartment. Here did the young ghost-hunter take up a position. . . .
When published in Bentley’s, the tale was the first installment of a series. The series was enjoyed enough to be reprinted in book form as The Ingoldsby Legends in 1840. And again in 1861. And in 1867. And in 1870 and in 1889 and in 1894 — and I think you see my point. It had a long life among readers. No doubt it played an important role in popularizing “ghost hunter” as a label for people who pursue spectral matters.