Electronic – High voltage Dickson charge pump switching

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.

I don't see anything dynamic here. No positive feedback, no bistable circuit.

There is no AC input (except maybe ripple at R1, where it does no good). There is no transistor switching. The transistors Q1 and Q2 are assembled as a two stage amplifier, rather than as a bistable oscillator - why? Perhaps R6 should be tied only to R7 and C2...

There are no rectifier diodes, which I expect to find among doubling capacitors. Perhaps R7 should be replaced with a diode?

The two transistors Q3 and Q4, below the LED, are connected in parallel - which is a distracting complication, rather than a true example of a LED constant current sink. Although biased (as shown) by a constant 2VDC supply so the base current is constant, this pair will not limit LED current using feedback. Perhaps if Q3's base was tied to the others emitter, and Q3's collector was tied to the others base...