Is it possible to install `tcpdump` on an Arduino 101?
You are better off with a Raspberry Pi or a Beaglebone board, as either of those are full-blown computers that are directly running Linux variants. You would find that the standard implementation of tcpdump would directly work on them (if it's not already included in the base operating system).

The Arduino Ethernet shield is basically a separate processor that implements some portions of the Ethernet stack and repackages it in a way that the simple micro controller in the Arduino can easily access. It will not provide low-level access to the OSI protocol stack. So I doubt that you would be able to get access to the full IP package contents.