I had the same issue with mine. After removing the starter, I found out the soleniod actually does 2 things. It engages the gear by tripping a lever and it provides 12v for the starter windings. I was unable to find parts or a rebuild shop. (every shop told me I had to buy a new starter at about 400.00
Since the gear was engaging (moving forward), I just installed a 12v automotive soleniod like used to be mounted on the firewall of older trucks (approx. 70's). The cost was about 15.00
Since I plan to repower soon, I just wanted to be able to make it through another season or two.
The only problem I have is occasionally the starter starts to turn before the gear is engaged. It kinda locks up, I just let off the key and try again. Usually within a few tries, it turns right over.
I don't know if this will help you, but it has worked well for me so far.