I bought a couple of the same Seimitsu PSL-30N-5W2 5V illuminated buttons.
I got the ‘smoke’ coloured ones.
The terminals are clearly labelled - switch, and LED + and -. They are 5V LEDs so I just cannibalized an old USB cable and soldered the power wires (red and black) directly to the button LED terminals. The white and green (data) wires I just snipped off, as they aren’t required.
The switch cables from the original buttons were attached to the switch terminals on the new buttons.