There doesn't exist a means through which a state could grant citizenship because the federal government grants citizenship.

Green cards are the next step below naturalized citizen. So you are looking to establish a process to become a green card holder, or lawful resident. That is, once again, a federal process.

To make a state originated green card would require an act of Congress and a constitutional amendment because granting citizenship or lawful resident status is a power granted to the federal government.

So basically no chance without a major change in voting, like a clean sweep of federal and mid-term elections with a supermajority Dem congress. That would not really result in a state originating green card, but would allow for mass naturalization of illegal immigrants once the appropriate appointments were made. Enacting a state originated green card at that point would just get challenged and would get struck down due to citizenship being granted to the federal government in the Constitution and affirmed by the Supreme Court in Chirac v Chirac.