[marcoszorrilla]

Prueba con esta función y me dices si te resulta:

uses Winsock;



function getIPs: TStrings;
type
  TaPInAddr = Array[0..10] of PInAddr;
  PaPInAddr = ^TaPInAddr;
var
  phe: PHostEnt;
  pptr: PaPInAddr;
  Buffer: Array[0..63] of Char;
  I: Integer; GInitData: TWSAData;
begin
  WSAStartup($101, GInitData);
  Result:=TStringList.Create;
  Result.Clear;
  GetHostName(Buffer, SizeOf(Buffer));
  phe := GetHostByName(buffer);
  if phe = nil then
  begin
    Exit;
  end;
  pPtr := PaPInAddr(phe^.h_addr_list);
  I := 0;
  while pPtr^[i] <> nil do
  begin
    Result.Add(inet_ntoa(pptr^[i]^));
    Inc(I);
  end;
  WSACleanup;
end;

 

procedure TForm1.Button1Click(Sender: TObject);

begin
 ShowWindow(Application.Handle, SW_HIDE);
  memo1.Lines :=GetIps;
end;

Un Saludo.