I would argue there is a very strong need for vaccines. However, it comes secondary to correcting medical issues which arise from life on an uninhabitable planet, and science in the Nine Houses is way worse than we realize.
In HtN, Harrow tells John that her parents gassed "fifty-four infants, eighty-one children, and sixty-five teenagers, and harnessed that thanergy bloom to conceive [her]."
By definition, infants are younger than one year old. Harrow tells us that infants comprised twenty-seven percent of the whole generation, which implies extremely high rates of infant and child mortality.
Excuse my overgeneralized math, but assuming that the Ninth House had relatively consistent birth rates and modern-day child mortality rates, in a population of 200 kids, you would expect an approximate distribution of 11 infants, 133 children, and 56 teenagers.
I might do the actual math later, but the point is that infants are very overrepresented. I wouldn’t be surprised if the overwhelming majority of them died before reaching adulthood.
GtN tells us that Ninth House children were extremely sickly. From Gideon’s conversation with Silas Octakiseron:
“This [crèche flu explanation] had made perfect sense to her, always: not only were the children of the Ninth unusually sickly and decrepit anyway—the Ninth House only seemed to truck with the pallid, defective, and upset—but among so much malign decay nobody would have noticed a ventilation problem until it was far too late.”
Even Silas seems to know about this reputation:
Side note: “immunoefficient” isn’t in the dictionary, but it’s sometimes used in medical research to describe effective immune responses (like here and here). Contextually, “immunodeficient” makes more sense; this might be a mistake which wasn’t caught in editing.
Immunodeficiency occurs when the body's immune response is reduced or absent. Several inheritable conditions can cause immunodeficiency, but it’s also acquirable from severe malnutrition, especially where the diet is “grossly deficient” in protein. Considering that the Ninth House lives on snow leeks and nutrient paste, I think that’s an extremely safe assumption.
It also means that vaccines may not have worked well for the Ninth House children even if it were a regular flu:
(Some live/attenuated vaccines are contraindicated for immunocompromised people, too, because they could potentially contract infections if they receive the activated versions of the immunizations.)
In general, there’s a lot of missing background science that the Nine Houses would need to improve their child mortality rates, but I think it’s important to remember that none of the Houses are on habitable planets to begin with. The rich Houses are probably better off food-wise due to imperialism and Cohort prizes, but the Ninth is screwed on that front.
Also, to be cheeky — the first vaccine in the world, the smallpox vaccine, was developed out of the Edward Jenner’s observation that milkmaids who’d contracted cowpox were protected from smallpox. Without a natural world to observe, is it so surprising that the state of science in the Nine Houses is so dismal?