First, it is legal to hunt deer in WA state with a 9mm, but......
The Hunting regulations state the following......
http://www.wdfw.wa.gov/wlm/game/hunter/handguns.htm"Hunter responsibility has always been a critical component of the hunting experience and now--more than ever--it is imperative that hunters make responsible choices when selecting a particular firearm or cartridge for handgun hunting. The recommendations below represent guidelines prepared for your consideration by hunter education program staff. It is now legal, for example, to use a 9mm handgun to hunt deer or elk. Such a decision, however, would be inappropriate since there is no way that the 9mm cartridge is sufficiently powerful for big game hunting. Even though the 9mm may now be legal, it is not a responsible choice for big game hunting. "
I handload both 9mm and 357 mag. I think that based on my past deer hunting that a 357 mag is marginal for deer. If I was certain, that I could stalk to within say 10 to 15 yards of a blacktail deer (they are quite small) and I had a +P 9mm load with the right bullet, I might be tempted if I was certain that I would not take a shot at a farther range.
I have meet a guy who likes to go out in the woods and "touch deer" with his hands. He is that good at stalking. Since you have indicated that you are a bow hunter, if you are that good, a 9mm might work and be a responsible choice.
Personally, for me, I am debating if I should hang up my 30-60 Mod 70 this year and try a Ruger Blackawk 357 Mag with a Red Dot scope on the top and some very heavy bullets with heavy powder loads for blacktail deer at slightly farther distances. I have not yet convinced myself that what I might do is "responsible." I intend to spend time at the range this summer thinking about using a 357 Mag.