Well, I am not sure if you application is for Internet or Intranet.
If it is intended for internet, I am afraid that would not be possible. Or I should say ASP.Net does not have a way to do it, you can always write some kind of ActiveX control to do this.
Now comes the intranet thingie. You can try this one:
Dim host As System.Net.IPHostEntry
host = System.Net.Dns.GetHostByAddress(Request.ServerVariables.Item("REMOTE_HOST"))
string strComputerName = host.HostName