I wonder if ignitors work depending on ballast output voltage? I am personally trying to look for ways to fire up a 18w SOX lamp on a US F20T12 preheat series choke designed for 120v 60hz supply. Will a European 18w SOX ignitor such as the Philips SX26 or the Venture PB018W255 ignitor work for this setup if a 18w SOX lamp runs at 57v 0.35a? I did try this setup with a European 35w SOX ignitor and the 18w lamp did not fire. I did find that my European 35w SOX ignitor did fire my 35w SOX lamp on my F40T12 preheat ballast.

They are.
The ignitor has to distinguish between lamp state: If it is not burning, it should provide ignition pulses. If it is burning, it should do nothing.
And the parallel ignitor has no other way than to sense the voltage across the lamp and start to fire, once the voltage correspond to the mains and stop once it correspond to the expected drop of a burning lamp.
Now the European ignitors are designed mainly to work on 230V mains, so 230V OCV (so about 320V peak), the burning lamp has about 57V. So the decission threshold could be anywhere between these two, with a reasonable margin to either limit.
So it depends, how the ignitor is designed.
Now you should remember, the makers tend to limit the number of part numbers for all their products, so the commercial SOX ignitors for 18W will most likely be designed for all all lamps sufficing with 230V OCV, so up to about 120V arc voltage. That would mean the threshold would likely be somewhere in the 180..240V range. And that means the 120V mains wont make it trigger.

However SOX ignitors may be a bit more complex, but generally are build of standard electronic components (no special devices like SIDACs used for MH or HPS), so by reverse engineering and adjusting some component values you should be able to tweak it so it will work with the 57V arc 18W lamp on 120V mains. And the Philips ones used to really be just a PCB snapped inside a plastic box, not that hard to open and repair or modify.

what components of tge Philips SX26 ignitor should I change out?

I would have to see the complete schematic - so you may open it and take detailed photos of the board and/or try to redraw it yourself...

The trigger element uses to be a ~30V diac, operating from a resistive divider sensing the lamp voltage. But needs to be identified there exactly.