from .NET 2 upwards the connection string can be stored in a specific node of the web.config file and not in appSettings as it was done before.
I am not sure about your question because if you try to access a database server located outside the network where you have deployed your ASP.NET application (so the hosting provider's internal network), it won't probably work, even in case your SQL Server is publicly available and open for connections from outside (which is bad, really, really bad...), it would also depend on the hosting company network settings.
How did you connect to the SQL Server during development and for debugging within the development machine?
There are info on how to connect to a SQL Server with a connection string in the web.config here: Using connection strings from web.config in ASP.NET v2.0 but not sure this solves your issue.