They do mention a manpower shortage.
However, really I doubt it takes all that many volunteers to crew Starfleet--a high estimate of 10,000 ships (at a high average of five hundred crew apiece) give a requirement of only 5 million line soldiers. Throw in another 5 million marines (if they even exist), and at a ten-to-one ratio of support-to-line troops, 50 million logistics officers, that's only 60 million in total, not even an order of magnitude greater than the combined numbers of the volunteer armies that exist on Earth, today. Given the factor of hundreds by which the Federation population outnumbers today's Earth, I don't find it too unbelievable that they'd have been able to generate 60 million, or even 120 or 240 million over the course of the war (to account of losses similar to those of the Soviet Union facing Germany). It's a miniscule fraction of the total population. At most 1%.
A far greater percentage of Britons volunteered to be be gunned down by Germans in the mud for the sake of Belgium; something close to that percentage of Americans are in voluntary service, even given rather unpopular wars abroad. I don't find it too hard to believe Starfleet could fill most of its gaps with volunteers, given the existential nature of the Dominion threat.
Plus, after the war started, the economic commissioners probably started hoarding antimatter for the starships instead of the public holodecks. Stripped of their right to holoporn, the citizens of the Federation suddenly, for the first time in centuries, felt the need to kill.
I do agree that it would have been a nice story to show some relatives or something, affected by the transition to a total war economy.