Where to put SQLite database file in Azure App Service?

In a Web App, your app is deployed to d:\home\site\wwwroot. This is the area where you may write files. As an example, the ghost deployment writes its SQLite database to d:\home\site\wwwroot\content\data\ghost.db. (easy to see this, if you open up the kudu console via yourapp.scm.azurewebsites.net):

This file area is shared amongst your web app instances. Similar to an SMB file share, but specific to web apps (and different than Azure's File Service).

The content under wwwroot is durable, unless you delete your app service. Scaling up/down impacts the amount of space available. (I have no idea what happens if you scale down and the smaller size has less disk space than what you're consuming already).

I would say the best location would be app_data folder in the site/wwwroot folder. Create the folder if it doesn't exist.

Web Apps can connect to storage accounts so you can in fact use blob storage and connect that to your web app. So in terms of learning more about it then you need to be looking at the appropriate page of documentation.

In your Web App settings you can then select which storage account to use. You can find this under Settings > Data Connections where you can select Storage from the drop down box.