Here in NY too, Byb. NYC used to have "well-baby" clinics, totally free, and no "needs" requirements. both of my kids got most of their inoculations, lead test, TB tine test and general physicals there. (my Mother-in Law at the time was a NYC PH service nurse, and worked in the Well baby clinic--so our "first visit" was as much to for her to show off her grandson as it was for her grandson to be weighed and meaured.)

Most of the well baby clinics have closed... now many hospitals have similar "wellness programs" to help kids get inoculations, etc.
as an aside, even for people who have health care insurance in US, it is often only for illness, not wellness, and normal check ups are not always "covered"-- a sick baby will be treated for free, but to get an inoculations there is a fee!