Not through the BCL, no. You can P/Invoke InternetGetCookie:
[DllImport("wininet.dll", CharSet=CharSet.Auto)]
private static extern bool InternetGetCookie(
string url,
string cookieName,
[Out] string cookieData,
ref int dataSize);
That third parameter should not use out, mind you - it should be marshaled as an [out] param using the OutAttribute only, otherwise the CLR will crash. See the Platform SDK for usage, but it's really pretty easy to do. To easily allocate a string of a certain size, you can use new string('\0', _size_) or declare those as StringBuilders and modify your source accordingly. Then you can parse and include these in your CookieContainer for use with the HttpWebRequest.
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