@Simon
"Grubbing" actually means to dig about or poke about (in soil or similar). An animal might grub about in the dirt looking for insects, worms or other grubs.
So money-grubbing is like someone digging or poking about in soil but digging about for money. So always on the lookout for money; poking about in the dirt for it.
Money grubbing and money grabbing have become synonymous really with the US leaning towards the former and British English towards the latter but they are interchangeable.