Folks have already pointed out the whole “multiple teams of sled dogs” thing.

The dogs also weren’t delivering vaccine. They were delivering diphtheria antitoxin, which is a treatment for diphtheria, not an inoculation. The kids were already sick.

There was technically a vaccine for diphtheria then (it was invented in 1923, two years before this epidemic), but it wouldn’t have saved anyone who’d already caught the infection.