It sounds like its a solenoid module. Here is what you can do step by step:check all electrical connections particularly the battery main connections these can easily corrode, in most cases that might be the problem Dismantle then wire brush inside and outapply some copper grease to each and refit Easy said but done but if you have some knowledge strip the starter and check the brushes etc. good electrical connection is key
I had a starting problem with my older model LT155. Apparently there was a consistant problem with the starter solenoid in the older models. Mine would hang up for a few seconds after I released the start on the ignition. It got to where it wouldn' start. I replaced the solenoid and the no-start problem persisted. A voltage meter showed the battery voltage went from 12.37 volts to under 6 volts when I hit start on the ignition. So the solenoid was working, but the starter was having a locked rotor problem. I took the starter to a repair service and the magnet in the starter was broken and jammed the starter motor so it wouldn't work. It was unrepairable so I picked up a new one and installed it. And it starts ok now. $15.50 for the solenoid and $179.00 for the new starter. (ouch) A word about the old starter solenoid. The old one is a 4-hole bolt pattern and the new ones are a 2-hole bolt pattern. They add a large fender washer to jury rig the new solenoid to the old bolt pattern. Is in the package with directions on how to modify it to work.
Neil
same problem as mine, check wires going in to solenoid switch they are calmp on wires inside a black plastic sheath on each side of the solenoid and loosen off also the starter where the key goes in gets gummed up so clean all the contacts you can get at, cheers roger