Actually, you probably can run tcpdump: consider the following
which tcpdump #NB: I ran this cmd as root to ensure I could find it
-rwxr-xr-x 1 root root 729772 Jul 13 2006 /usr/sbin/tcpdump
so it can be run by anyone IF you supply the absolute/complete path.
Whether you have perms to monitor a given port/device is a different qn; if you haven't, ask your SysAdmin.