Green’s Dictionary of Slang

1. an eater; thus heavy grubber, an enthusiastic, if unmannered, eater.

1819		P. Egan Key to the Picture of the Fancy going to a Fight 16: [T]he Lower Orders of Society [...] are good grubbers.
1830		Egan Finish to the Adventures of Tom and Jerry (1889) 221: Blacky punished the steaks, swallowed all the potatoes, and took the lining out of a quart of porter, like winking. ‘Well done,’ cries Tom, ‘by heaven! you are a fine grubber.’.
1838		‘Paul Pry’ Oddities of London Life I 235: Ven I axes her at dinner for a bit more, she chucks ony von tater at me, and a bit of meat vot aint of no use to sitch a ‘heavy grubber’ as I am .
1843		R.S. Surtees Handley Cross (1854) 152: You are light i’ the girth and don’t look like a great grubber.
1861		T. Hughes Tom Brown at Oxford (1880) 57: I like to see a fellow an honest grubber at breakfast and dinner; but you’ve always got your nose in the manger.
1902		Bulletin (Sydney) 4 Oct. 13/2: Dined yesterday at a sixpenny restaurant. A crowd of wharf-lumpers and country yahoos staying at the place, two blackfellows and a blind man at table. [...] Another caught her skirts when passing, while nearly every paltry ‘sixpenny grubber’ indulged in idiotic familiarities, and yelled her Christian name with ‘dear’ affixed from the end of the room whenever he wanted his ‘hot water’ replenished.

2. (US) the mouth.

1889

Times-Democrat (New Orleans, LA) 9 July 3/6: Prize Ring Slang [...] ‘kisser,’ ‘grubber,’ ‘trap,’ ‘whistler,’ ‘ivory-box,’ the mouth.