I'm taking my kids camping in the Uintas (Kamas region, Mirror lake, etc.) this weekend and I'm hoping for a random dispersed site instead of a camground. My kids are still a little too small backpack into anything, so it needs to be accessible by car. We can hump loads 100 yards or so if needed.
Was just out there a for a few days. If you see a non-marked dirt road, go ahead and check it out. Generally there are "sites" there, but it's technically BLM land so you can set up shop wherever. I found solace down past Ruth Lake just beyond the Sulfur campground. Bring bug spray. Mosquitoes were gnarly. We were tucked back in the woods a bit, but there were fire pits and flat spaces for tents. We also had a momma moose and her calf trot right through our site. The Uintas are rad! It'll be a bit of a drive to get back to Mirror Lake, but Ruth Lake is definitely worth checking out.
A couple of other options if you can't find anything off the main highway are heading down either Soapstone Basin road or Murdock Basin road. Both have quite a bit of dispersed camping spots. The downside however is that it's going to be further away from most of the climbing so you sacrifice some convenience, and those areas are popular with the ATV crowd so it's probably not going to be incredibly peaceful. Good luck and have a good trip!