A little further looking into it.
Daniel Hess invented a vacuum cleaner in 1860 that didn't use bags, it bubbled the air and dust through tanks of water to remove the dust. this is, as far as I can see the first "bagless vacuum cleaner" in the strict sense of the term.
To take the term bagless vacuum cleaner to mean a cyclonic cleaner then it seems to be a Mr. E.H. Yonkers that invented the first one in 1928, initially sold as "Health-Mor Uprights" and later as Filter Queen. (see my link up there^)
James Dyson invented the dual cyclone vacuum cleaner sometime in the late 70's, he took a patent out on the idea in 1986 and the first dyson vacuum was released in 1993. (I say invented, he more adapted it from commercial dust extraction systems)