I think the hard part is most of this is higher altitude so snow is you biggest issue. Most of the higher elevation won’t be accessible until like June or July. That might be the case for a lot of that area. I want to say that around those lakes is like 7k-10k ft.
These trails aren't too far from Park City (Strawberry OHV Trail System - ATV & Dirt Bike Trails Heber, Utah | Rider Destinations). I have ridden these in winter on my snowmobile, but have not ridden these in spring/summer.
The area that Jdance posted is called the 'High Uintas' area off the Mirror Lake highway (150). Mid-May there is always snow remaining, and the paved 150 road may not even be open (check with Forest Ranger office in PC). In summer, it is a really awesome area. The Heber City & Strawberry areas are lower and likely OK as we haven't had as deep of snowpack as normal years. Hope this helps.