I want to get involved in emergency preparedness. What organizations, if any, are there in the United States that support disaster recovery and emergency management?
The level to which amateur radio operators participate in emergency management varies between municipalities. ARES is the most well known organization, it is managed by emergency managers in each ARRL section. To join, submit the application form to your section emergency manager. You need not be an ARRL member to join.
RACES is managed by state and local government - it's separate to but associated with ARES. Basically, ARES operates at most incidents and events. RACES assumes control when an emergency is declared, but it should be the same group of volunteers in both cases. The best explanation I've seen of the differences is on the ARRL site.
Of course, you can also be useful as an amateur radio operator without being part of one of those organizations. Local volunteer fire and ambulance services, CERT teams, and even offices of emergency management need communications, and being able to work on their frequencies as well as on the ham bands can make you a valuable asset.
In the United States, major NGOs maintain amateur radio capability; specifically, the American Red Cross http://www.redcross.org/find-your-local-chapter plus the Salvation Army with its North American SATRN network http://www.satern.org .
There's also military affiliated organizations. MARS http://www.netcom.army.mil/mars http://www.afmars.org/mars1.shtml http://www.navymars.org/national/mission.htm , the Civil Air Patrol http://www.gocivilairpatrol.com/ and the USCG Auxiliary http://cgaux.org/about.php all need communicators.