Yes, you can program the Series Six with a standard PC. There are two versions of Logicmaster Six software, the parallel version, which is what you probably have and a serial port version.
The parallel version requires what GE calls a WSI (Workstation Interface) card. The WSI card in your Workmaster computer can be installed in a different computer. The original Workmaster computer with the tiny screen was an IBM XT machine. The WSI card in that model installs in any 8-bit ISA expansion slot. It has a daughter board so it physically consumes two full size slots. The Workmaster II is an IBM PS2, and it uses the IBM micro channel backplane instead the ISA or EISA backplane. If that's what you have, then your chances of finding a replacement computer are not so good. The connection between the WSI card and the PLC uses a cable with 37-pin d-shell connectors and connects to the I/O Control module in the CPU rack.
The serial version of Logicmaster Six communicates via the COM1 or COM2 port of your computer, so it doesn't require the WSI card. However, it requires a Communications Control Module in the PLC. If the Communications Control Module is located in the slot to the left of the combined memory module in the CPU rack, you can communicate without having to add any ladder logic to your program to support the comms. If the CCM module has to be located in an I/O slot, then the program has to include a special rung of ladder logic to enable the comms.
That's a quick overview. Let me know exactly what you're working with and I can fill in the details. Remember, Logicmaster Six is a DOS application. Your best bet for a replacement computer is still going to be something old. I keep a 386 machine around just for servicing these old units.