for example Hawaii has quarantines for dogs and cats entering the state
Those are strictly precautionary quarantines. US Federal law and most other nations do the same thing for animals traveling internationally.
In terms of livestock animal quarantines, even for human communicable diseases, I'm not sure it is ever the CDC, and is instead USDA. For example BSE (Mad Cow Disease);
It depends on whether the disease is transmissible to humans from live animals (anthrax).
For BSE, the cause has never been positively identified and evidence of transmissiblity to humans is weak at best, and even if it is, it's transmissible only by consumption through the food supply. That left BSE in the USDA's jurisdiction.
Small outbreaks tend to be handled by state/local officials.
If an outbreak of a disease transmissible from live animals to humans happens on an interstate scale, actual orders for livestock quarantines may go through the USDA, but the CDC will take the primary roll because of the potential need to quarantine people who have been exposed to infected livestock.