There is a small parking lot that a few trailers can fit in at the gate. My in-laws live in Randolph and when I was there last week there was no snow in Randolph or Woodruff. The gate really isn't up that high so I bet if you unloaded there you would have a fair amount of asphalt before you hit snow.
IMHO the riding up any of the canyons along Bear Lake (Logan, Fish Haven, Bloomington, Paris) is far superior to the riding at Monte. Plus a lot closer.