Even kill shelters don't usually kill puppies. They are rediculously easy to rehome. It's usually the older dogs/pit mixes/dob mixes that aren't as cute/cuddly that bear the brunt of kill shelter ethics. 15 puppies are going to be quite a responsibility. That being said, if you are convinced you must, this is the way it works here:
You do not need a prescription for the vaccines, from the feedstore it comes all together, single dose w/ needle included.
These vaccines do not need to be administered by a vet, you can do it yourself and it is not difficult.
In most states the only vaccine that must be given by a vet is Rabies, and I'm guessing these pups are probably too young for that anyway.
It's not legal in all areas to give meds yourself, but in our area it is.
Hope this helps, and good luck. I do hope that none of your puppies get sick. Is there a way you can vaccinate them before you bring them home? And please remember even then immunity takes time to develope.