Is it possible to measure TDR without a specialized device?
I'm looking to determine the feasibility of implementing a TDR
  utilizing existing network infrastructure.


The Ethernet cable's capacitance is directly proportional to its length. So a network analyzer could, I suppose, perform an out-of-band in-band cable length measurement by injecting a fast rise time u(t) pulse into one end of the cable, providing a resistance of known value at the other ("measurement ") end of the cable, and measuring the RC network's rise time to to determine the cable's length. This technique would allow for auto-ranging (selecting different resistance values under software control), and relatively low-cost hardware could perform this type of measurement (e.g., a sufficiently fast microcontroller with comparator inputs and counter-timer circuitry). This would not be a "true" TDR, but it could, I think, measure a cable's length with adequate resolution and accuracy.

Would it be feasible to retrofit an existing network infrastructure with this kind of cable length measurement system? In my opinion, no; it's not feasible. Firstly, you must design this retrofitted cable length measurement system in such a way that it does not degrade the 100/1000 Mb/s signaling on the cable. Good luck with that. Secondly, the switching/routing software that's executing on the iron inside the switch/router is several orders of magnitude slower than the time of flight of the electrical signals on the cable (at least within a local network segment this would be the case). So there's no point trying determine the time-of-flight differences between cables A and B when those time values are on the order a few nanoseconds, considering the software running on the switch/router requires around one millisecond (&lt;-ballpark estimate) to process the incoming packet and decide which port to relay the packet out of, or to repackage the information for the next hop along the routing path.