Can somebody help me find the right kind of connectors for these illuminated buttons?
The ones in the photos are "spade lugs" and they are pretty common in automotive shops. However closer perusal of the pins shows that two of the pins are thick and two are thin. Getting spade lugs to hold on these will be an exercise in futility.
They really look like these pins should be soldered. Push a wire through the hole, fold it back on itself, solder and then cover with heatshrink. Put whatever plug you like on the other end of the wire.
If you Google for that part number on the switch, you might find there's a purpose-built socket for it but I don't like your chances.