System.FormatException: Invalid length for a Base-64 char array or string.

You need to look at exactly what the web request is returning: it isn't just a Base64 string, there is presumably some packaging around it.
Use the debugger, or log the raw data to a file and examine it: there is stuff I think you need to remove - but we can't do that for you as we have no access to your code or the URL...

You should read an array of chars from the stream and then to finalize your stream reader before to do the conversion.

byte[] bDoccontent = null;
HttpWebRequest hwrProg = (HttpWebRequest)WebRequest.Create(URL);
HttpWebResponse hwrpProg = (HttpWebResponse)hwrProg.GetResponse();
StreamReader sr = new StreamReader(hwrpProg.GetResponseStream(), Encoding.ASCII);
char[] base64CharArray = new char[sr.BaseStream.Length];//New
sr.Read(base64CharArray, 0, (int)sr.BaseStream.Length); //New
sr.Close(); //New
string base64String = new string(base64CharArray); //New
bDoccontent = Convert.FromBase64String(base64String );//New
Response.Clear();
Response.ContentType = "Application/pdf";
Page.Response.AddHeader("Content-Disposition", "inline;filename=wei.pdf");
Response.AddHeader("content-length", bDoccontent.Length.ToString());
Response.BinaryWrite(bDoccontent);
Response.Flush();
Response.Close();