Exotic animal owners licensed by the USDA are currently unable to access the website or APHIS information. This governing agency provides guidelines and rules for breeders, brokers, exhibitors and dealers. While not crucial, some licensees may be hampered by inability to check status, regulations and contact officers through email during the government shutdown.
With an undetermined length of time of inaccessibility, owners/licensees should conduct their business as usual, adhering to regulations and animal welfare standards. A government shutdown doesn't release any owner from standards of care or regulations.
Contacting your local inspector by telephone is likely the best avenue for immediate contact, but waiting it out is an option for non-crucial information.