Green’s Dictionary of Slang

(orig. RN) an infantryman.

1868		G.J. Whyte-Melville White Rose 57: I shouldn’t like to be a ‘Grabby’ [...] I’d rather be a private in the cavalry than an officer in the regiment of feet!
1900		Bulletin (Sydney) 8 Dec. 28/2: On foot they journeyed never, ‘Grabbies’ or ‘toe-soldiers’ might ‘mud-crush,’ but not ‘gentlemen-hussars’ with red gold belted around their waists.
1909		J. Ware Passing Eng. of the Victorian Era 146/1: Grabbies (Country). Infantry. Probably disguised grubbier, from the evident fact that the infantry are not out of the mud as are the cavalry.
1919		‘Etienne’ Strange Tales from Fleet 123: ‘Do you mean the grabby’s dinner-party we gave?’ inquired the recumbentfirst lieutenant .