On my DOS machine, the PC is only there to hold/edit data and to transmit it to the central controller of the PnP. The PC just has an expansion card installed with a big fat centronix cable out the back which (on mine) goes into a serial port breakout board. There are 2 serial cables going from that breakout board to the central controller, but during picking and placement, nothing is sent from the PC to the PnP machine, you can turn off the PC after you send the data. And actually, you can even leave the PC off when you fire up the PnP, hit start, and it will continue placing boards with the last program that was loaded in.
In my machine, the lower left side of the cabinet (when facing the machine) contains the PC, and in the back is the air input and the transformer. The lower right side front contains the main controller, vision cards, etc. The rear is the servo drivers.
So at least in my machine, there is not much in the way of connections between the PC and vision ISA card. Is your vision card installed inside the PC itself?
The PC shutting down or crashing with a card installed or not, and/or when trying to connect to the network isn't normal... if that was my PC I'd be starting with a fresh format and re-install of Windows, and expecting to have to replace the motherboard.
Let us know how you make out.