Hi, Mike. I’m going to try to answer, but somebody may find it necessary to correct me.
I haven’t been in that area recently, but the Horseshoe Meadows road is typically closed many miles from the trailheads and campgrounds in the winter. You could theoretically hike up the road, but that just isn’t practical. I assume that this is why they don’t show any permits.
Many years ago, I did a lot of winter climbing. They never closed the dirt roads that lead to the unpaved trailheads along the eastern sierra. Taboose Pass, Shepherd Pass, Red Lake (Split Mountain), etc. were always open. They used to plow all the way to the Big Pine Creek trailheads, so those were always accessible. They only plowed part way to North Lake, South Lake, and Sabrina, so you had to hike up the road to get to the trailheads.