Is there any way to store data on the Overwolf platform, as a database? For using as shared data on a network-based game. Or we have to take a web service/database for this?
You can store the relevant data in your app’s directory.
You can read here how to get the app directory using the overwolf.paths API.
But this is based on the local user machine, not on the Overwolf server.
For shared info between OW users, you will need an external DB.