Is it possible to have a website shut down because it carries
objectionable or ofensive material ie animal cruelty

If hte contentonthe website is actually illgeal, then yes you can. 
Normally the agencies to investigate these are the fbi and cia.  If
you think the website is not just obejctionable, but illegal, you
should report it to your local police deparment and ask them what to
do.  Some will take action from there for you, some will make you go
through the proper channels yourself.

Actually, it's virtually impossible.

Without knowing the actual site I can't say much more about the
subject other than to list the multitude of reasons it is difficult
and how the Web site owner could counter every effort, including by
simply moving hosting to another country which has no laws at all
regarding the Internet.