Not my post but follow these steps for diagnosing the problem.
You really need a multimeter to diagnose it correctly. If you are not even getting a click when you hit the button, then try the simple things first before you tear anything apart.
All these should be true in this order:
-12v at the battery while pressing the start button (meter red probe on + and black probe on -)
If you don't bad battery
-12v at the battery while pressing the start button (red probe on battery + and black probe on a CLEAN, BARE metal spot on the motor...not a bolt head,not a fitting...a CLEAN BARE spot of the casting of the motor even if you have to make a spot)
12v while pressing start button -->ok
immediate drop in voltage when the start button is pressed--->bad ground
-12v at both solenoid posts while pressing the start button (red probe on one post at a time, black probe on battery - )
While pushing the start button, if you get 12v on both posts and get no cranking then your problem is the starter or cable ends at the starter
If you get 12v on one post but not the other then your problem is in the S/S switch circuit.
S/S switch circuit is real easy to diagnose with the ebox apart. With the box open:
-There is a little red wire spliced into your battery cable that connects to your fuse holder. You should have 12v inside the side of the fuse holder that connects to that wire (red probe inside the fuse holder, black probe on the ground post that the starter relay is grounded to)
No 12V-->bad splice or connection to the battery cable, or bad ebox ground from stator
Have 12V-->ok
-The brown wire that connects to your starter solenoid, unplug this wire and test the end coming from your s/s switch (red probe on brown wire connector and black probe on ground post that the solenoid is grounded to)
While pressing the start button:
12v on brown wire coming from switch-->bad relay or relay ground
No 12V on brown wire-->your problem is in the S/S switch or any associated connection starting from the red wire spliced into your battery cable and ending at the brown wire that plugs into your solenoid.
If nothing simple solves it then follow that order, it will take everything out of the equation. As long as you are testing the connections correctly and there are no goofy or ghetto fab connections someone made or altered in your ski you'll find the problem.