What does it mean when my 4 week old toy puddle gags?

It could be a hairball... or simply them standing up too fast... my chihuahua does that... half the time its because she sticks her whole face in the water bowl to take a sip of water. Keep in mind that small animals have small organs.

That doesn't sound good at all. I would get it to the vet this week. No responsible breeder would let you have a 4 week old. That is only when you start to wean them. It needed to stay with its mother and litter for a couple more weeks. If the breeder is letting the puppies go too early, they may have neglected the puppies health too. Let the vet examine it, and plan a shot program.