Originally, back in the 1700s, a bug-hunter was UK slang for an upholsterer. This was apparently under the theory that they dislodged bugs in the course of their work.
In the 1800s the meaning changed, coming to refer to someone who robbed from sousers -- either through pickpocketing, swindling, or simply rolling drunks. Again, the meaning was probably strongly influenced by the possibility of finding vermin during the act of liberating the contents of the marks' pockets.
During the same period, more respectable dictionaries were reporting this phrase as a synonym for entomologist. The term bugologist was also used, although not in formal contexts.
In modern times, a bug hunter is someone who looks for bugs in computer code, usually for profit. The term has been bolstered by Google's use, although they do not use it as a formal term; bughunter.withgoogle.com links to the rather more dully named 'Vulnerability Rewards Program'.