Outside of the state LE draw hunts (archery= one week axis deer, one week mouflon) on state leased land (a big chunk of the island) you can buy a "public private" permit that's fairly reasonable and hunt some real good areas with good numbers of animals. There is a bunch of archery only areas.
And all this could change tomorrow too. Or has already yesterday. Between the state and the property owners.... many things have and are being done they promised they would never do. As in many places and many states..... the inmates are or have taken over the asylum.
That puts a different spin on it. There are several places on Lanai that private folks do everything from a true B&B to rent a bed in a converted garage. We do the garage route, you get a place to sleep and a hot (mostly) shower, hotplate, microwave, etc. Many of these also offer vehicles to rent. Neither are as cheap as you'd think they should be but not bad. Lanai is an expensive place just to exist. You'll pay $100 a day for a truck you often can see the road go by through the floorboards Gas right now is $5 or 6$ a gallon and (normally) only one gas station. Closes at 9pm. I have a list somewhere of folks who rent places, but if you go that route be on your best behavior, your most polite. You are literally a guest in someone else's home in a rural tightly knit community. You don't want to be the poster boy for 'flippin haole......... some of these relationships take years..... decades, to develop. Big mouth Jimmy Big Time or Rowdy might just..... go missing.....
But these places are not what most wives are thinking of WRT a trip to Hawaii.... to say the least. Although if willing and fiscally able it can be literally a 5 star trip/hunt. It's only money (lots and lots of it) but you can't take it with ya! Lanai is an awesome place. But it moves at it's own pace. Does things it's own ways. Settle in and enjoy it. Every rich jerk that comes along and buys the island seems to forget that real quickly, mostly when their mainland bean counters and lawyers they put in place take over.....
Maui is MUCH more logistically friendly. And very good hunting, they have everything Lanai has but mouflon and have feral goats and pigs that Lanai (theoretically) doesn't have. You can stay is a real nice somewhat reasonable resort or condo and get in day hunts here and there. Several private guides/outfitters available with access to private land, On Maui that's key as there is VERY little public land to hunt. Rodney has a good rep, I'd suggest Pat Fisher as well. Those are the main two guys on Maui.
Molokai has several private outfitters as well. That's still a bit of a rustic rural trip for most wives. Big Isle is logistically friendly like Maui, has mouflon, feral sheep, goats and pigs. But (theoretically) no axis deer. Those are my hot button honestly and only available in Maui County (Maui, Molokai, Lanai) Kauai has some great hunting, pigs, goats and even blacktail deer. Those deer I understand are a tough hunt. Jungle deer.
if you're staying on Oahu, may the Lord have mercy on your soul..... pretty much like every other urban hellhole on the planet.... IMO.... always too many rats in a cage....
I hunted with https://www.facebook.com/mosmasicampo Mos Masicampo for deer. We were on a holiday and stayed at the Four Seasons and he just picked me up at the hotel. It was a holiday and I was able to sneak in a few days of hunting. He is a taxidermist and a good bowhunter. I don't know all the other HI options or what you are interested in but we were doing a sun/beach holiday and it worked just worked out to hunt a few mornings and then meet up with my wife and other friends who were holidaying there. Good luck.
