John,

1-battery OK? Do you have a 12V test lamp you could put across the battery and see if it dims or goes out when trying to start? Or a voltmeter?

2- starter solinoid could be bad, it could (A) not be making the connection from the battery to the starter. (B) not moving the gear up (bendix) (C) click you hear could be just the safety interlock clicking due to a bad seat/brake or other interlock switch.

If you can link a diagram of the electrics it would be easier to go further and suggest some tests before you start replacing things at random.

You could if you know how jump the solinoid and see if the starter spins but be aware not to short anything out, unless your familiar with how to do that I would suggest you do not.

Starting with the ground battery terminal, disassemble every connection point in the starter circuit - use emory cloth/sandpaper to clean metal connections to bright surface. M/W, remove battery -take to a shop that can LOAD test same.
If still a prob, and if unit has a briggs engine, it may need a valve adjustment. Another thing to look at is if it has a gravity feed fuel tank, pull plug and turn engine over several revs to ck for raw fuel in cylinder. Also pull dipstick and ck for raw fuel in crankcase oil - Change oil right away if so.Then fix carb float needle and install/use inline fuel shutoff valve.