Since PbCl2 is insoluble in water, how is it that lead leached from pipes due to Cl- reaction can become ingested?
Lead(II) chloride is not the sitting-at-the-bottom-like-a-brick type of insoluble. Rather it is the well-ma~ybe-I’ll-dissolve-a-little type of insoluble. So enough lead chloride will be dissolved — as Ivan mentioned, enough to poison somebody. Its solubility is given as $10.8~\mathrm{g/l}$ on Wikipedia. There are much less soluble salts like mercury(I) chloride ($2~\mathrm{g/l}$), barium sulfate ($2.4 \cdot 10^{-3}~\mathrm{g/l}$) or silver(I) iodide ($3 \cdot 10^{-6}~\mathrm{g/l}$). Note that for mercury(II) sulfide a solubility is not even given, it is merely termed insoluble (and is the sitting-at-the-bottom-like-a-brick type of an insoluble compound).

All values were taken from Wikipedia and are given at $20~\mathrm{^\circ C}$.