In the input offset current diagram [like Eggleston Fig. 6.12], what is drawing current?

The op-amp inputs draw ``bias'' current, i.e., there is current $I_B^-$ flowing into the op-amp inverting ($-$) input and current $I_B^+$ flowing into the op-amp non-inverting ($+$) input. This is in contradiction to the op-amp rule that no current flows into the inputs, but that rule is an idealization. Inside the 741 op-amp (and similar), there are actually bipolar transistors, and the op-amp inputs actually go right into bipolar transistor bases (see the schematic on p. 7 of the 741 data sheet). So the bias currents are the transistor base currents, which are small but non-zero.

Op-amps made of FETs will typically have much smaller input bias currents.