As well as the above possible causes listed above by 'the hobbler', it might also be a problem with the starter switch that is mounted on top of the starter motor - this is what is mechanically operated when you pull the start lever inside the car.
If you click on the 'Downloads' section at the top of the page, then select '500 Classic', you can download both a workshop manual and an owner's handbook for the 500. In this workshop manual, Chapter 11 - Electrical, covers the overhaul of the starter motor and shows all the parts.
As regards starter motor spares, if you input Fiat 500 Classic spare parts into your search engine, a list of spares suppliers should come up. Afaik, no one does a kit of parts for the starter motor, but many do individual parts - I think most necessary parts are still available. I'd suggest stripping your starter motor for inspection before ordering any parts - it might need very little, possibly just clean the commutator and fit new carbon brushes as 'the hobbler' has already suggested.
Al.