Can a living organism run on electricity?
No; the problem is, as you pointed out, that no organism will manage to multiply, grow or even sustain itself without absorbing matter to create new cells and fill metabolic losses.

Even photoautotrophic organisms which get energy from light (which is in fact an E-M wave, so pretty close to electricity) collect matter from the environment -- plants for instance seem to grow out of nothing, but in fact gather significant part of their mass from atmospheric CO_2.