The programming software is called Proficy Machine edition. Communication with that model PLC is serial, not Ethernet, so you will need a serial port on the computer that runs Proficy. That could be a USB/serial converter or a real serial port on an older computer. You can use the 15-pin RS485 port that the HMI is connected to or you can use the RJ45 (serial RS232, not Ethernet) port behind the door to the left of the HMI cable.
Your local distributor can provide the software and the proper cable. Most GEXPRO offices are authorized to sell the product line.
Another alternative would be to try to find a nearby integrator with GE experience who can come in and do the job for you. Based on your description of what you want to do, I doubt if it would take more than a few hours for someone with GE experience. Assuming you're in Florence Kentucky your proximity to GE Aircraft Engines in Evendale Ohio and GE Appliance Park in Louisville means there should be more than a few people familiar with GE PLCs nearby. GE's corporate policy was to specify OEMs of equipment for their factories use GE PLCs.