Here you will find how to create an Azure SQL Database, 24 x 7, and starts with approximately USD$5/month (basic tier). Make sure to add your current Internet IP address to the whitelist of Azure SQL Database firewall so your application can connect from your location. Use this documentation to configure Azure SQL firewall.
Download and install SQL Server 2012 Native Client from here.
Modify the connection string for your application:
Private Function GetSqlServerConnectionString() As String
GetSqlServerConnectionString = "Driver={SQL Server Native Client 11.0};" _
& "Server=tcp:YourAzureSQLServerName.database.windows.net,1433;" _
& "Database=YourDatabaseName;" _
& "Uid=YourAzureSQLAdminUser;" _
& "Pwd={xxxxxxxx};" _
& "Encrypt=yes;" _
& "Connection Timeout=30;"
End Function