We have always taken Jeeps and 4WD pickups, not yet an HMMWV.
The
White Rim Road has a maximum vehicle width of 8' and height of 9'6". You do need an overnight permit; I suggest 2-3 nights to truly enjoy it and cannot think of a reason why a well maintained and outfitted HMMWV driven by a capable driver would have any trouble traversing the road.
It is beautiful with plenty of opportunities for vistas (like your photo of the Shafer Trail), hikes, and photo ops. We are looking forward to our next trip.
Here is one from above looking down upon the White Rim above the Colorado River. If you zoom in you can see the WRR snaking around Lathrop Canyon.
A couple not far from the start of the WRR entering from the Potash Road and Shafer Trail intersection. Musselman Arch and a Colorado River overlook.
Let us know when you go and post some photos!