Official CDC guidance:
Make plans to get vaccinated early in fall, before flu season begins. CDC recommends that people get a flu vaccine by the end of October. However, getting vaccinated early (for example, in July or August) is likely to be associated with reduced protection against flu infection later in the flu season, particularly among older adults. Vaccination should continue to be offered throughout the flu season, even into January or later.
If you have not already done so you should get vaccinated for Covid-19 first, says my own doctor. And then do your next required vaccines.
I had MMR early this year, and Pfizer Covid vaccines in March/April. I plan to get the influenza vaccine in September.
(Getting vaccinated for MMR is weird for an old idiot like me. While I had measles and mumps as a kid, I never had rubella and no doctor had ever asked or advised me about getting the MMR vaccine until I asked my own doctor about this early this year. Everyone I know around my age, even if they had all of MM and R, was subsequently vaccinated, sometimes even twice!)