Twice in my life I have needed to get a cat from Point A in central time to Point B somewhere in California. Handling all the vaccinations and associated paperwork was (in one case) annoying or (in the other) the person who actually owned the cat not remotely organized enough to have records.
It turns out some domestic airlines will permit you to fly with a pet on-board in a soft-side case under the seat where one might normally put a bag. This may or may not work for you, depending on how patient and large your dog is and how big your budget is.
The person to whom you replied is trying to import an animal into a foreign country. Doing that, even for pet-in-cabin and service animals, requires a certificate from the USDA. The same process as domestic flights won't work in this circumstance, unfortunately.
My particular issue is that I'm trying to arrange to transport my dog across country lines on an airplane. But the county I'm flying to requires my vet to complete a vaccination certificate which then has to be stamped by the USDA APHIS office (normally no big deal). But of course this is currently impossible because USDA APHIS is shut down and doesn't even answer their phones. It's not clear if any alternate path exists to get around this.