Send the ComboBox a CB_SHOWDROPDOWN message, passing a (true) in the WParam parameter to drop the list down, or a 0 (false) to hide it.

procedure TForm .Button Click(Sender: TObject);
begin
  SendMessage(ComboBox .Handle,
              CB_SHOWDROPDOWN,
               ,
              0);
  ComboBox .SetFocus;
end;