Answer: Scabies caused by Sarcoptes scabiei (a.k.a. the "itch" mite)
This is a very remarkable case in that the mites are so numerous and that they were still alive when we received them. This degree of parasitism is consistent with Crusted or "Norwegian" scabies, occasionally seen in the elderly and individuals with a compromised or suppressed immune system. Patients with crusted scabies typically exhibit thick crusts of skin that contain thousands of mites. What is shown here is one of these crusts.
Some viewers commented that these organisms bore a resemblance to ticks; this is not surprising, since ticks and mites are both arachnids in the subclass Acari. They are differentiated by size (ticks are generally larger), and have a toothed hypostome and Haller's organ which ticks lack.
Thanks to everyone who wrote in on this case!