You are unlikely to get enough current from a normal 9V battery for it to work for long, if at all.
It would probably need some capacitors across the DC input, to provide the current pulses without too much voltage drop - eg. a reasonable size electrolytic in parallel with a plastic film cap to handle high frequency edges.
Those so-called ZVS oscillators are not very efficient to start with as both transistors are on for part of each half cycle (so not very much not "zero voltage switching"), and whatever current you draw at the output will need 100x more from the battery, not allowing for those inefficiencies.
If it works properly with an adequate power source, you could get rather more than 100x input voltage, as the transformer primary should be resonating, boosting the drive.
Also note that air breakdown is approx. 3KV per millimetre.
Make sure you use high speed diodes such as fast recovery or Schottky types; normal rectifiers turn off rather slowly and are not suitable for high frequency circuits.