There is some hunting on the Kitsap Peninsula, but mostly on private land wit hfirearm restrictions, but he will be only a short distance from the Olympic Peninsula and the Olympic National Forest http://www.fs.fed.us/r6/olympic/. This area is in the Fish & Wildlifes Region 6 http://wdfw.wa.gov/reg/region6.htm
This area is full of opportunity for deer, bear and elk. Not to mention all the fishing, clamming and crabbing!

Good luck!

I am also in the Navy stationed an hour and a half away from Bremerton. Does your son use Bow or Rifle. My first hunting experience was last year and it was successful. Went to the good spot and bagged a good one. I'll be going east this year north of Winthrop and try Mule. I'll be scouting September and Season opens usually for modern rifle is October. I felt left off when i got here in Washington from Texas for hunting but i just scouted and ask until i found a good spot. I feel what he feels. Nobody wanted to show me their spot that's why i went scouting and paid off. What's his e-mail?