Not impossible
Alaska Airlines serves Kodiak on a schedule and they recently changed their jumpseat policy (They'll take as many jumpseaters as they have open seats in the back).
Unfortunately Alaska limits their cockpit jumpseat to Northwest, Peninsula Airways and ERA Aviation (all three codeshare with AS).
ERA also has several flights a day but I don't know what their policy is.
Peninsula operates flights 135 on the island but has no 121 "mainline" to Anchorage.
For the cargo carriers try Empire (FedEx feeder) and Northern Air Cargo.
The jumpseat policy at NAC is currently under revision.
Your best chance at jumpseating will be if you're employed by a scheduled 121 operator.
The crews at ERA, Empire and NAC are all very relaxed.
Clearing security with Alaska might be a problem.
Good luck!