A money-grubbing person is one who greedily seeks to obtain money at every opportunity. The phrasal adjective derives from the noun money-grubber, which uses the verb grub in the sense of to dig or search.
Money-grubbing dates from the early 19th century.1 It is possibly the source of money-grabbing, a slightly less common adjective that made its first appearances later in the 19th century.2 The two words have the same meaning. And while it is tempting to say money-grabbing is probably just a mishearing of money-grubbing, there is no clear proof of this. Both words make literal sense, and both are well-established in English.
In current U.S. news publications, money-grubbing is far more common than money-grabbing. The inverse is true in British publications, where money-grabbing is preferred.
Shed the belief that networking is a ploy to get ahead by money-grubbing soulless glad-handlers. [Washington Post]
[A]ccording to Trivers, psychoanalysis is nothing more than a money-grabbing hoax. [Guardian]
[T]hat Buckingham/Nicks melodrama is still played out on the money-grabbing reunion shows. [Stuff.co.nz]
The Scottish Government can then claim to have helped Scots businesses against those evil money-grabbing tyrants in London. [Scotsman]