Here's the fix.
Copy these two libraries into their respective locations for the client sites, as follows:
cp /lib/x86_64-linux-gnu/libnss_files.so.2 /var/www/clients/client#/web#/lib/x86_64-linux-gnu/
cp /lib/x86_64-linux-gnu/libnss_dns.so.2 /var/www/clients/client#/web#/lib/x86_64-linux-gnu/
(NOTE: The location of these files may vary depending upon your distribution, but the names of the libraries will be the same. You can use strace with ping to find out where it's looking for them.)
There are a lot of useful command line tools that are missing from the default client jail, such as ping, wget, dig, nslookup, etc. Not sure why these were left out. Maybe I did something wrong during the installation.