The local.settings.json is only used for local testing. It's not even exported to azure.
You need to create a connection string in your application settings.
In Azure Functions - click Platform features and then Configuration.
Set the connection string
A function app hosts the execution of your functions in Azure. As a best security practice, store connection strings and other secrets in your function app settings. Using application settings prevents accidental disclosure of the connection string with your code. You can access app settings for your function app right from Visual Studio.
You must have previously published your app to Azure. If you haven't already done so, Publish your function app to Azure.
In Solution Explorer, right-click the function app project and choose Publish > Manage application settings.... Select Add setting, in New app setting name, type sqldb_connection, and select OK.
Application settings for the function app.
In the new sqldb_connection setting, paste the connection string you copied in the previous section into the Local field and replace {your_username} and {your_password} placeholders with real values. Select Insert value from local to copy the updated value into the Remote field, and then select OK.
Add SQL connection string setting.
The connection strings are stored encrypted in Azure (Remote). To prevent leaking secrets, the local.settings.json project file (Local) should be excluded from source control, such as by using a .gitignore file.
https://docs.microsoft.com/en-us/azure/azure-functions/functions-scenario-database-table-cleanup