I have a paternal aunt that appears to have been mentally retarded, and institutionalized in the 1940s in Pennsylvania. I had thought this woman died around 1945 or so, but it turns out that she died in 1989, in the institution in central Pennsylvania that she apparently had been sent to.
In her case, her gravestone was listed in Findagrave.com, which then was listed on Ancestry.com. Also, the institution where she was sent must have informed Social Security of her death, because there is also a Social Security index listing on Ancestry.com.
Perhaps that is a way to find out what happened to your relative, by searching for a Social Security claim application or in their death index?
In Pennsylvania, there is a form that a person can fill out, to get the medical files of a deceased institutionalized relative, but you do have to show some close relationship to the person. I haven't tried to do this. Maybe the state of New York has such a form? In PA, the form is not sent to the instituion directly, but to the state department of health (IIRC), or whichever agency oversees institutions.
What a surprise to us when this relative showed up on Ancestry.com. I had never heard of this poor person. I don't think her siblings (my father and sisters) were ever informed of her death. And I'm certain my mother never heard of her, because my mother would have endlessly complained about this (she endlessly complained about my father's mentally ill first cousin).