When Titus conquered Jerusalem in AD 70 he also found the Jewish law stone tablets. Flavius Josephus mentions this in “The Jewish War, Book VII, Chapter 5). These tablets can be seen on the Titus arch as well.
It is unknown whether these tablets are the ones Moses received during the exodus. But at least we know two such tablets existed in AD 70 in Rome. According to my present knowledge the trace ends there.
A search does not make sense in my opinion. First, much more information is needed to know where to start looking. Second, even if stone tablets are found there is no way to know whether they are identical to ones conquered by Titus.
(C) 2006-2011 Thorsten Straub, www.biblical-finds.com