Ok, not the sharpest when it comes to electronics but it looks to me as if the output sinks. If that is the case then you could use the relay but it would have to be different. What you would have to do is have 24v going to the relays coil and the other side of the coil would connect to each of the fault outputs on your drive, you would then need to use the NC contact of the relay rather than the NO that you seem to be showing.
So what would happen is the coil is fed by 24v but it would not connect to 0v until one of the drives faulted. So in a Non-Fault position the 24v connected to the relays contacts would pass through the NC contact and onto the E-Stop string. If one drive faulted then the coil would get powered and the relays contact would open and thus the E-Stop string would be opened thus signalling an e-stop.
Hood