NAMI is a non-profit organization, has no rules anyone has to follow.
Unless you have your Medicaid under a 1115 waiver, you should qualify for what you are looking for -- does not mean particular clinics will necessarily take Medicaid but there should be some that do; call the State Health department and see who takes your insurance in your area:
https://www.dhs.state.il.us/page.aspx?item=29728
https://store.samhsa.gov/shin/conten...05-0202-IL.pdf