Where are you hosting this? If locally/internally on a physical computer or VM (which it looks like you are) and not some cloud service like Azure or AWS, then you need to grant the user running the AppPool read/write access to the /App_Data directory.
If running in a cloud service, then make sure you're not storing those files in blob storage or storage off of the website.
Check what David mentions, *Are there any files created in the folder: CMS\App_Data\CMSModules\SmartSearch[your index name]*. If there are then you know it's not permissions.
Lastly, check the scheduled tasks. There are 2 tasks regarding search indexes
- Optimize search indexes
- Execute search tasks
Make sure they are enabled.