Flu vaccines are not part of the childhood vaccine program. The main reason for the flu vaccine shortage is unpredictable demand and inability to stockpile vaccines from year to year to smooth out supply. Because flu strains change yearly, manufacturer must destroy unused doses at the end of the year, which eats into profits. For example, in the 10 years before last year, manufacturers typically destroyed 10 million unused doses of vaccines yearly. In other words, there were plenty of vaccines.
Last year though, early publicity predicting an especially severe flu seasons caused unexpectedly high demand. This year the loss of one manufacturer has cut into supplies and the publicity over a shortage seems to be creating an increase in demand, which nobody predicted.