SQLCLR TVF that calls web service is getting error 401: Unauthorized
A couple basic SQL Server checks: Is your database set to trustworthy?  Is the assembly set to allow UNSAFE execution?

If that seems to be working, there are a couple of tutorials out on the web that might help.  If none of the tutorials out there help, then it sounds like a permissions issue.  (Unfortunately, that gets tricky, as this thread discussing Kerberos versus NTLM authentication shows.)

One other check--as Maumen mentioned--make sure that you have anonymous access setup for IIS.  

Ultimately, though, it sounds like something you want to avoid, if you can.  The performance and memory implications, the difficulty getting all the accounts working, and decreasing all the security settings would make me a bit nervous.