I believe United States Postal Service doesn't allow shipment of loaded ammo or anything explosive. Other carriers are good to go but I don't think USPS allows it.
Also, not that anyone cares but I'll vent and rant for a bit, that BS tax Wa. charges when you purchase firearms out of state is just that, total and complete BS!!! Luckily, the FFL I use only charges $20 to do transfers unlike most that charge $35+