... Operating two motors from one switch is no problem.
It is in this case. That contactor is rated for 5HP 230V, he already has that maxed out. Adding another 1/2HP 230V single phase motor is a significant additional load on that unit, the contacts will not take it for long. Correct fusing will be trickier too.
But otherwise, I agree it's more likely the voltage is wrong. Start there before jettisoning it if you are planning on just getting another separate contactor for the 1/2HP motor. If you still want to do them together with one contactor you will need to replace it with a larger contactor anyway, so no point in trying to fix it.
To do that you will need one rated for 7-1/2HP (the next size up). But also, the only viable way to run them both from one contactor is if the 1/2HP motor has integral thermal OL protection built-in, information that will be on the motor nameplate. If the nameplate doesn't SPECIFICALLY say that it is protected, you must assume it is not and use an external OL relay. The wording is usually something like "Thermally Protected", but might just be abbreviated as "TP", or something with similar meaning. Otherwise you will need to add a second OL relay to that contactor in parallel to the one for the 5HP motor and wire the aux contacts in series so that if either motor overloads, it drops out the contactor. Rather than go through all that, it just might be easier to get another small starter for the 1/2HP.
If you do need a new coil, that contactor is not likely available any longer. Rockwell, at that time that was made, did not yet own Allen-Bradley so they used electrical controls for their machinery from other sources. When they bought A-B they dumped all of their other arrangements for controls. But Rockwell is no longer in the machinery business anyway*, so there isn't anyone there to find out who actually made that contactor in order to be able to order a coil.
There is a "new" company using the name Rockwell for tooling, they are in no way related. The "new" Rockwell (RWDirect Inc., division of Positec) is a Chinese manufacturer who is surreptitiously using the Rockwell name to capitalize on the brand recognition. The "old" Rockwell International division that made tools had been merged with Delta and when Rockwell Intl. got out of the tool business, that division changed their name back to Delta International Machinery. Delta, I've heard, does not maintain records on the older Rockwell branded stuff however. I don't know that for sure, that's just what I was told at a local Delta shop when I tried to get some parts for an old table saw.
PS: I punched that contactor part # into Delta's site and got this:
Delta – Service Net >> Use 438- 1- 16- 199
The one they say to replace it with is also not available any more.