Non-profit organizations and the Local, State, and Federal Governments cannot do it all. People who can take care of themselves lighten the load on available public resources.
During any active emergency and during the recovery period following a disaster there is a good chance you will be on your own for quite some time. An event that was supposed to last a few hours may last days, weeks, or months. I don’t mean to sound preachy but you may want to give some serious thought beforehand on how to survive a variety of situations. Stock some supplies and make plans for different possibilities. Have plans to bug-out as well as plans to shelter in place. You may find it best to follow instructions issued by emergency agencies. For example, should the authorities say to evacuate due to an approaching Category-5 Hurricane with 30 ft. tidal surge, you may want to pack up and make the run to higher ground.
Include plans and provisions to save your pets. While agency-run disaster shelters may have room for you, I doubt that they will accommodate your critters.