From what I have been told the USDA regulates it on a Federal level - and there can be additional state regulations.
Here's a link I was sent:
USDA
These are the regulations on chinchillas:
(3) The following persons are exempt from the licensing requirements
under section 2 or section 3 of the Act:
(i) Retail
pet stores which sell nondangerous, pet-type
animals, such as dogs, cats, birds, rabbits, hamsters, guinea pigs,
gophers, domestic ferrets, chinchilla, rats, and mice, for pets, at
retail only: Provided, That, Anyone wholesaling any animals, selling
any animals for research or exhibition, or selling any wild, exotic,
or nonpet animals retail, must have a license;
http://www.aphis.usda.gov/ac/cfr/9cfr2.html