Placing oxygen molecule into electric field
Yes, an oxygen molecule (much like any other molecule) can be broken by electric field alone.
No, it won't be broken into separate oxygen atoms, precisely because the atoms are neutral and the bond between them non-polar, like you said. Instead, it would be broken into $\ce{O2+}$ and an electron.
This is not important anyway, since it would require electric field much stronger than those present in air ionizers.


What happens in air under voltage, then? Well, air always contains some ions from whatever sources (cosmic rays, background radiation, etc). As you apply voltage, your positive and negative ions start speeding up in the opposite directions. Once the voltage is high enough, they accelerate to the point that they might break neutral molecules by bumping into them. In doing so, they produce more ions, and those do the same, so pretty soon there is a breakdown going on. You see a blinding spark, the fuses blow up, the voltage shuts off, and the ions start to recombine into molecules. As luck would have it, sometimes they combine into $\ce{O3}$ - that's an important source of ozone both in nature and in the lab.

In fact, the situation is much more complicated. Gas discharge does not have to be that dramatic. It comes in various forms, some of them pretty controllable.