Just to elaborate for anyone unfamiliar - these laptops don't have a joystick port on the laptop itself but there is one on the optional port expander/docking station/whatever, so the joystick signals apparently go through there.
Can you post a clear photo of the connector? It might help somebody identify it, if in fact it's an industry standard connector. Below I've tried to post a picture of what the mating side on the dock looks like.

If it's impossible to source, I suppose you could get a dock for it and remove the connector (unless you want to just use the whole assembly as-is), but that's expensive and I have no idea how difficult it would be to remove. I used to have one until a few months ago. The way the dock is designed maybe there's an intermediate connection that just unplugs - but I never took it apart so I have no idea. If it's directly soldered to a PCB then maybe the pins are long enough they can just be cut off instead of desoldering.

I looked through my old photos from when I sold my dock but none of them had a clear picture of the connector. This is the best I could come up with:

Photo attached. The port is about 70mm long and there are 200 pins.

From your photo it looks like the mating part is just an exposed PCB so it might be possible to design a replacement with a 3D printed housing. In a way that actually sounds easier than soldering a 200 pin connector.

Getting hold of the dock itself would be the preferred option but after a few months searching I've yet to come across one - building a small breakout board with joystick and MIDI would get me 90% of what I want (the only missing thing would be networking, which I can live without).