Not sure what the "best" organization might be, but the bigger national organizations are all out there (HSUS, Best Friends). Austin Pets Alive is one of the shelters that has been taking in a lot of animals from the affected areas.
Or if you want a local organization, Fur-ever Fosters on Mishawaka Ave. is serving as the local drop-off location for donations for Austin Pets Alive. More information is on their facebook pages: https://www.facebook.com/fureverfostersanimalrescue/