To hunt unclassified wildlife, you need a valid big game or small game license. You can hunt unclassified wildlife year round with no bag limit and with any weapon. For instance, you can legally hunt unclassified wildlife with an air rifle.
Attached below is the Small Game, Furbearers, and Unclassified Species section (Pages 132-133) from the 2009-2015 Game Management Plan available at http://wdfw.wa.gov/conservation/game/.
6th of December, 2013