Let's collect all our witty comebacks to this common silly question into a FAQ. Thanks to all who have contributed. Here's what we have so far, keep them coming, limit 250 submissions per person:

"The legs and both tails of Pembroke Welsh corgis are docked at birth."
"Horrible accident with the lawnmower. The vets were able to re-attach the feet."

"This is a Sheplar, a cross between a German Shepherd and a caterpillar."
"This is a Lapland Lemming Herder. They don't really need much in the way of legs."
"... Sawed-Off Shepherd, the latest in family and home protection."
"... Deutschundpig: cross between a German shepherd, a dachshund, and a guinea pig."
"... Bernese Airliner Dogs. They're bred to fit in overhead compartments or under the seats."
"... Collie/otter crossbreed, a.k.a. Cotterpin."
"Ottoman Shepherd Dog... a favorite breed of the Ottoman Empire's ruling class. After a hard day of ruling the people and conquering new lands, there was nothing like coming home to your tent, sitting down in your easy chair, and putting your tired feet up on an Ottoman."
"When she was a puppy, we made the mistake of putting her into the microwave to dry her off after a bath."
"Shortcake is her favorite treat."
"Ever heard of the Texas Chainsaw Massacre?"
"One leg was too long, so the surgeon shortened it -- a little too much -- so then she had to shorten another... try leveling a 4-legged chair sometime... it would be easy if they had only 3 legs"
"She was born with six. We had the vet remove the extra pair."
"He's still just a puppy. This breed lives to perhaps 150 years, and isn't full-grown until about 40 years old."
"Television stunted his growth."
"It's this dreadful virus, Appendagioma abbrevialis; it's highly contagious and pretty soon everybody's dog is going to look like this. They THINK it's not transmissible to humans, but nobody's really sure..."
"We got her from a breeder near the Hanford Nuclear Reservation. Her legs just never grew properly. Her claws glow in the dark, it's really cool."
"PLEASE. Don't talk about it in front of her. She's awfully self-conscious about it."
"I'm very proud of them - they chose to be leg-donors, to less fortunate dogs."
"These aren't dogs, they're giant mice."
"You know Seattle... too much Starbucks stunts you."
"You see that yummy looking thighs? One day, I caught my husband chewing on one of the dog's legs! It was half gone, so we had to chew the rest off to balance it off. The dog didn't mind. In fact, she helped herself to one of them. :) Chomp chomp!"
"What happens in Vegas - stays in Vegas."
"Skydiving accident."
"The cats ganged up on him so he can't reach them as easily."
"These dogs were genetically engineered for people too lazy to walk their dogs."
"We wore them off by too much walking."
"She's not short; she's standing in deep mud."
"I was scared he would jump off the balcony."
"Oh yes, my blind brother tried nail clipping."
"Gangsters bred these short sentry dogs to guard the coke, cash and guns stashed underneath the beds and furniture."
"Don't try to clip claws with garden shears."
"I'm short, so I got a shorter dog to make me look bigger."
"I pulled on his ears when he was a puppy, and that pulled in his legs. That's why his ears are so big."
"She's been on short rations."
"They don't need long legs. When food is involved, they're the world's fastest dogs, even on 6" legs."
"We didn't read the label.  It says "AIR DRY ONLY."
"He didn't jump the jump rope fast enough."
"My parents have joint custody and my dad got the shorter half."
"I told him not to stick his tongue in the electrical socket."

"They grow back each time I cut them off.  It's like mowing the lawn..."

"Her legs stopped growing when they reached the ground."

I NEVER expected anybody to come up with a straight answer, but this will shut them up every time (thanks to David and Shari):
"An Expressed Fibroblast Growth Factor 4 (Fgf4) Retrogene Is Associated with Breed-Defining Chondrodysplasia in Domestic Dogs (including dachshunds, corgis, basset hounds and at least 16 other breeds of dogs). (Science DOI: 10.1126/science.1173275), if you must know".
It's true, no poop.

Science 325:995-998, 21 August 2009 (with a corgi sighting on p.918!).
SCIENCE online

Finally, of course:
"Legs? What legs?"