Do students and faculty comprise any State University in the US? Oxford University as a corporation grants membership to the matriculated students who are currently pursuing a degree. However, in the US there seems to be a tendency for University legal business to be handled by a corporation of University Trustees which is a legal entity separate from the University as such.

Sometimes the Trustees are said by statute to be "included" in the "body corporate" of a University, which seems to make two different bodies corporate, one being the Trustees acting in their special name, and the second being the University. However, it isn't made clear who else might be included in the University besides Trustees or for that matter whether  the term "University" can also refer to the buildings or district under jurisdiction of the Trustees.

I would be interested to know if there is any State University which makes its students or faculty members of the University. Please cite the relevant laws.
No, there's nothing like Oxford's or Cambridge's legal structure in the United States, where students and faculty are members of the University.

For historical interest, the first university -- University of Bologna - 1088 -- was a universitas scholarium, i.e. a "university of scholars (students)".  It was a collective that had bargaining rights with the city and hired and paid their professors -- paying them after each lecture. (!)