An older model Snapper riding lawn mower suddenly cut out during operation, is this the solenoid?

Not the best question, but the battery might not be the best condition, however, but it seems the starter motor (Bendix) gear is not throwing out to turn the engine. Remove the starter motor and take off and clean the Bendix gear on the shaft and give little a light oil on the shaft that will allow free movement the Bendix gear. Replace the starter motor and start again. If the Bendix gear is too worn then replace it with a new cheap Bendix gear unit; they are about two bucks from Lows. As an after thought, you might need to replace the spring, too.