I will say this initially in order to get dealt with, and then rejected. There most likely is a Ground wire physically available. That means there always is 120 VAC at the current switch point. If you were to measure from the electrical panel Hot side (not the lamp connection Hot) you will get 120 volts. And that can be used to continuously power the power supply. But this is not permissible under electrical code rules. It would be the easiest solution, but as I said it is not permissible. Constructing a device (non-approved) and connecting it to an electrical system is likely not permissible as well.
The most safe way to deal with this problem is to run the ESP off of batteries and change them when required. Of course this is inconvenient. Replacing the current wiring to add a neutral with at the switch point would be the right thing to do. But that would likely require removing and then replacing wall paneling.
With the schematic flash_os drew it seems that his idea was to power the ESP from the voltage at the switch when the lamp was turned off. There are a couple of problems with this. When the switch is closed the power source for the ESP is gone. This can be dealt with by using batteries to power the ESP and the intermittent supply could charge the batteries when the light is off. This might work if the lamp was an incandescent bulb but with LED bulbs the problem is more difficult. With incandescent the cold/off bulb presents a low resistance and the minor current through it (to power the electronics) will likely not cause any visible light from the bulb.
The normal LED on current is lower and the off state (dark) would likely not allow us to steal current through it for powering out circuit. The LED lamp is not a linear device and it will not likely give us the required voltage to the power supply as the incandescent bulb would.
All of this is why radio controlled switch replacement devices (Wemo) require a neutral wire connection. The best solution is to add a neutral wire following the wiring code rules. The easiest solution is to use the box ground as the neutral with but this is not permissible.
Whatever you do, please add a fuse in series with any circuit you use connecting to power supply for the ESP. As I said, a battery powered ESP is one solution. I would try and use a solid state relay (optically isolated triac circuit) to switch the lamp power if using battery power. This would require less current on the ESP side. This may or may not work well with some LED bulbs. There may still be flickering (light output) when the solid state relay is in the off condition. I have observed this on some bulbs that we have tested at work. We make a product using a triac to control and dim banks of LED lights. Single lamps tend to be a problem.
A different option is to locate the WiFi switch at the lamp rather than at the switch. This way you have continuous power for the ESP. This is likely your best option. A conventional relay could be used. It could be in series with the current switch or in parallel, depending on how you want it to work. The current switch always on if a series connection is used.