It isn't completely clear to me, but it seems to be this meaning of 'at someone' - 'Piggy and Jack endlessly
nagging at him'. Where 'nag' means 'to annoy, usually by complaining about faults'.
It's a common way to express this idea, although I usually hear it as 'get at someone', which is why I said it isn't completely clear if this is the meaning in this instance. Some examples where it means 'nagging':
My mum's been getting at me all morning about not cleaning my room.
His wife's constantly at him about not putting down the toilet seat when he's finished.
You get the idea