Welcome to the forum, Whidbey has lots and lots of deer, so you're in great shape there. If you are military, you mau have an in at NAS Whidbey for at least limited access. Otherwise, you're going to have to knock on doors, so to speak.
There are several members of this forum that hunt that island, though they may be currently inactive currently. Later in the summer, as thoughts turn towards the pending season, you might get further assistance from those people in specific Whidbey information. In the mean time, I'd do some parcel searches for large chunks of land as potential places to pursue permission, Island County Assessor's Office maintains a GIS searchable map that allows you to identify potential targets of interest and current owner's mailing addresses. Farmland and forestlands are your primary targets.
Don't ignore the rest of the local islands as well as the San Juan Islands, which are also loaded with deer. Several have public land that is open to hunting. Getting to the island becomes the major logistical problem. Also, Port Townsend and the Olympic Peninsula are a 45 minute ferry ride away. There is no end to the hunting opportunities in that direction.
Good luck!
