Depends on the time period. In the Middle Republic, the answer was no. Not even the auxiliary soldier himself got citizenship regularly. We hear of anecdotes of exceptional auxiliary soldiers being granted citizenship for their deeds by generals (for instance, Marius; see Cicero Balb. 46; Valerius Max. 5.2.8). This practice continued into the Imperial period, but again ONLY for auxiliary soldiers who had distinguished themselves as exceptional. This award of citizenship had nothing to do with any wife or kids, take note. If the guy died, and that death wasn't the pinnacle of heroic sacrifice, then the family of the deceased were out of luck. And note that being left destitute due to war was incredibly common even in the 20th century, much less in the ancient period.

From the time of the emperor Claudius, that changed: auxiliary soldiers "of good character" (so again, only the "good" ones) were awarded the right of conubium, which granted citizenship to them and to any informal marriages the soldier might have. The kids were then automatically citizens (presumably only the ones born after the right of conubium was granted). These awards were regularly recorded on bronze documents called diplomata, some of which survive.

For reasons we don't understand, from the time of the Emperor Trajan onward, there was another change: from that time, only the kids of the auxiliary were given citizenship who were born after the soldier's discharge (so any kids he fathered while in the service remained illegitimate).