Ideal Op Amp and its Input Voltage
I'm wondering how exactly the electric-potential energy from the EMF gets "transferred" across the resistor and to another section of the wire if there is no movement of electrons?


Imagine if there was a potential difference between the two sides of the resistor. 

Then a current would flow through the resistor, until the voltages equalized on the two sides of the resistor.

If it helps, also remember there is a small parasitic capacitance from the node labelled \$v_p\$ to any other nearby conductive objects (which we usually model as just being connected to the ground node). So when you first connect the 1 V battery, current does flow through the resistor, but only until those parasitic capacitors are charged up to the point that the potential is equal and current stops flowing through the resistor.