Is it possible to set up the realm URL, claim types, etc for azure ACS without editing the web.config? Can you set up these required elements programmatically somehow?
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Yes, FedUtil does this. It's a utility that comes with the Windows Identity Foundation (WIF) SDK and you can invoke it from visual studio.
http://msdn.microsoft.com/en-us/library/ee517285.aspx
http://www.microsoft.com/download/en/details.aspx?displaylang=en&id=4451
EDIT: I may have misunderstood your question. FedUtil is a utility that configures your web.config for you. If instead you want to configure your application in code, that's also possible. The WIF documentation on MSDN should demonstrate how to do this:
http://msdn.microsoft.com/en-us/library/ee766446.aspx