The proper way to use connection strings on Azure Websites is to add "debug" connection strings to your web.config
file (and by "debug" it can be a local db/storage or any string that is safe to share - empty string).
On the Azure portal go to your Azure website --> CONFIGURE tab and under connection strings sections add your actual connection strings with the same names as used in your web.config
file, there the connection string are saved as encrypted strings.
The website code will get the proper connection string you set in the Azure portal.