Sheep hunting with ATVs?
MTV
It can be done, but there are very few spots where ATV access will get you close to sheep. DCUA is one, but the drawing permit application window closed in mid-December. I know of a couple others in general season areas, but...there's a reason those areas are quiet.
The reality is that sheep are struggling statewide. A few winters of rain on snow icing events have taken a toll. There are still sheep in the hills, but the number of legal rams is way down in multiple ranges.
To the OP - you have to understand that even asking for general areas is like asking for another man's firstborn daughter. Especially on an open internet forum. There are relatively few areas where a guy hunting off the road system can have a reasonable chance of success, and even fewer in recent years based on the trend in sheep populations. The handful of places on the road system that hold sheep are closely guarded by those who know about them, and for good reason.
That said, it's not really rocket science. Figure out where sheep live (which, thankfully, is in most ranges in Alaska). Study the regulations and figure out where there are open seasons. Then, if at all possible, spend some time in there in the summer. Climb mountains and figure out where you're seeing sheep. Scouting is key. If you can find sheep in July, there is a decent chance they'll be around in August.
And yes, ask for help. Just understand that it's more likely to come in hushed tones in private conversations over a beer rather than on an open forum. There are sheep out there. Of all the sheep I've taken, only one was from a fly-out hunt. The road system can be productive. But it often will take spending a lot of time in the hills, a lot of scouting, a lot of conversations, and going into valleys that others might overlook. Don't hesitate to take a hike into a place where others don't go. ATV access...not likely to get you a ram from the trail, but hey, it might not be a bad place to start hiking from.
