Maybe, early pre-diagnosis aspects of eventual Alzheimer's (such as undiagnosed mild cognitive impairments) makes people less interested in, and diligent in pursuing, flu vaccinations. (In other words: the causality goes the other way.)
But also, people with active family/social lives are both more likely to be reminded to get a flu vaccine, and less likely to develop Alzheimer's.
And further: more conscientious people seek out vaccines. More conscientious people get Alzheimer's at a much lower rate. For example, this study - https://jamanetwork.com/journals/jamapsychiatry/fullarticle/... – suggests those in the 90th percentile of conscientiousness have 89% lower rates of Alzheimer's diagnoses than those in the 10th percentile of conscientiousness.
While a mechanism from vax-induced immunity, and/or fewer actual infections, to less dementia is plausible, these other well-known correlates/confounders seem a likely explanation for these results, until/unless controlled-for.