How does one obtain connectors for game controllers to put on a PCB if e.g our design takes input from a PS1 Dual-Shock controller?
I've done exactly that for a university project. I wanted to interface to a NES controller, so I had to find some connectors. I ended up buying a splitter adaptor, which gave me four connectors to play with. I then measured the dimensions and created a PCB footprint.



Buying parts for a PS2 controller may be a bit easier, as it is not as old as NES. Just look for a broken PS2 or even some kind of adaptors and salvage connectors from them.

Here's an adapter for PS2: