You cannot pull the 24VDC from the PF40 terminals, unless you like buying new PF40 drives a lot. That internal power supply is capable of feeding it's own input transistors, not much else else. Maximum output current is 100mA, but you still can't use it all without compromising your inputs.
Also, assuming you are in the US somewhere by virtue of you discussing 110V control, and given that the breakers used in that (Siemens SQ23) are not legal to use in the US, the chances are high that this was designed and built in some other country, where most likely that is NOT a 110V coil, but rather a 220V coil, on a machine designed to be fed with 380V 50Hz where the Line to Neutral voltage is 220V. If it were modified to supposedly work on a US 230V 3 phase system and they put in a 110V coil, someone did not understand the differences here. 230V 3 phase here is not Wye, it is always Delta, so there is no L-N 110V inherently in the design. There is a POSSIBILITY that you have what is called a 120/240V 3 phase 4 wire system, also called a "High Leg" or sometimes a "Red Leg" 230V delta system. In that system, one leg of the 230V delta transformer, usually phase A-C, is center tapped to give you 120V from A or C to Neutral. But if you are saying that you have no Neutral feed, then you don't have that. So your only choices are to 1) get a 230V 60Hz coil for that relay and run 230V to your control devices, or 2) get a separate 230V power supply that gives you 24VDC output for that circuit. I would go for the 24VDC power supply.
Either way, you will have to change the coil on that relay, and that's the other bad news. That relay has been out of production and support by Siemens for well over a decade, actually longer than the PF40 has been around, which means someone cobbled this together with old parts (or that it originally had a different VFD). So your only hope of finding a different coil voltage for that is going to be E-Bay or other surplus parts dealers. If it were me, I'd just replace the entire relay, it will likely cost you less.
"Will work for (the memory of) salami"
