Most emergency shelters won't accept pets on-site for public health reasons. As such, organizations such as FEMA and the Humane Society recommend research to become familiar with pet-friendly hotel options before disaster strikes. Consider programming the numbers of pet-friendly hotels and motels within a 100-mile radius into your phone. Make sure to clarify any necessary deposits required in addition to the nightly rate, as well as any restrictions on size, breed, type, or quantity of pets, in advance.
As a backup, look into veterinarian offices or animal hospitals in your area that might be able to provide fostering or assistance. Try to only rely on these as a last resort, as these places often become inundated during widespread emergencies.