How would I know if someone is ward of the state due to no family members?

https://www.agingcare.com/questions/my-mother-already-lost-her-memory-quickly-and-she-has-no-will-no-estate-planning-no-power-of-attorne-466293.htm

Your first question concerned your Mother. Who are you referring to in this question?

One does not automatically become a ward of the state/county due to no family members. Many people have PoAs who are not relatives because they have no relative, and some people use attorneys for PoA. Guardianship by the state for an adult happens when there is no PoA or other person with legally recognized authority for that person AND that person has been deemed incapable of making decisions in their own best interests. If a person has no relatives but has a PoA, or has no PoA but is not incapacitated, they would not be a ward of the state at that point. Three things need to happen: 1) a person is incapacitated; 2) the person has no PoA; 3) no one else steps up to pursue guardianship through the courts. That's when the govt steps in.

Based on how hard it was to find an answer online for my state, and having experienced my stepFIL becoming a ward of the county, I would guess this is not public information. My stepFIL's guardian (Lutheran Social Services) pulled down the iron curtain on all his information once they had control and didn't share anything with us unless we had a compelling reason (his wife was still alive and we were caring for her, so they were somewhat flexible).

Is this a member of your family? Why do you need this information? Is the person you need this information about competent, or in some sort of need of something? I myself am uncertain whether or not a ward of the state or a ward of the court is publicly recorded, but I would imagine it might be. You perhaps can check with your public records office and ask this question by calling.
I hope others have better information for you than I.