You can trace hardware that exposes trace functionality usually via a debug port of some kind. Many chips have trace functionality in their production design, but no debug connector is physically present in off-the-shelf boards (to reduce manufacturing cost). You can usually physically modify the board to get access to this functionality which is routinely done when porting software to a new chip/board.
Trace functionality comes in two major flavors, control flow trace and memory trace. Control flow trace only records control flow, so the contents of memory are unknown which is not very useful for your desired use case. Memory trace records memory accesses, so the contents of memory are known. Unfortunately, memory trace is very resource intensive, so most systems that support trace only implement control flow trace. As far as I am aware, it is very unlikely that any desktop or server CPU has memory trace.
The major manufacturers of trace probes and solutions that I know of are Green Hills Software, Lauterbach, and Segger.
