Is it possible to construct a voltage doubler that both powers an opamp and is driven by it?
As shown you have a perpetuum mobile, none of which work. You are using the opamp as the energy source for itself and that dog won't hunt.

You can do what you want if you use external switches driven by the opamp - 2 x bipolar transistors or MOSFETS. The opamp then supplies only the drive energy to turn the "switches" on and off. 

Lots of ideas here but

An opamp version of the circuit below would do what you want.
The opamp (here = 555) can be powered from the output - initially its Vdd is two diode drops below Vcc_in but this rises as the circuit bootstraps its own supply. 

Circuit from here





An alternative is to build a simple boost converter - oscillator (opamp or whatever) inductor, jellybean BJT transistor, a few resistors, diode, caps.Zener clamp output to regulate. Choose inductor size, frequency to suit desired power level.

Various examples here

including circuit below from here - good design notes on that page.