I would like to know if I can run a 35w Sox lamp on a 55w Sox ignitor because I have a Sox cu phosco p567, and it has a 55w ignitor but it can only have a 35w lamp. The ignitor only says 55w and doesn’t say 35w on it. I just don’t want to risk damaging any of the gear In the lantern, or the lamp.

Check the lamp arc voltages, if the 35W has it lower than the 55W, then it should be OK. But if the lamp arc voltage would be higher than the ignitor is designed for, it may not be able to properly recognize when it ignites so continue pulsing indefinitely, overheating or wearing it off.

The lamp works on the 55w ignitor, I just don’t want to damage any of the gear or the lamp. I don’t know how to check the arc voltage of the lamp.

If I recal well, both 35 and 55W are using the same nominal arc current, so the 35W has definitely lower arc voltage.
So operating a 35W using "55W" ignitor is perfectly safe, only the opposite (a 55W lamp on "35W only" rated ignitor) may be problematic.

In my experience, most lamps and gear can survive some mild misuse, at least when used just occasionally. Real world conditions may not always be optimal either.