This is still to be heard in Northern England, Scotland and Ireland, and most of the works in which it is to be found are by authors from those regions. Writers from those countries sometimes export it to a wider audience:
There are no smiles, only expressions of naked hostility aimed at the Leinster players as they retreat behind their line for the conversion: John O'Neill, with clenched jaw and jutting chin, and Foley, ball under his right oxter, shoulders back as he releases a triumphant roar.
Sunday Times, 23 Apr. 2006.
It means the armpit, though it can be used more widely to refer to the underside of the upper arm or the fold of the arm when bent against the body, or even the armhole of a jacket.
To judge from the way Robert Louis Stevenson used it in Catriona in 1893, it can have a wider figurative sense: “I’ll confess I would be blythe [happy] to have you at my oxter, and I think you would be none the worse of having me at yours.” George Macdonald Fraser almost made it sound rude in his Flashman and the Mountain of Light of 1990: “A lackey serving the folk in the gallery put a beaker in my hand. What with brandy and funk I was parched as a camel’s oxter, so I drank it straight off”.
The word is from Old English oxta, which has related forms in some modern Germanic languages. It appears to be linked to Latin axilla in the same sense, a diminutive of ala, the wing of a bird, and so is a distant cousin of aisle.