Democratic Party of United States v. Wisconsin ex rel. La Follette (1981 case about party nominations)
Someone tried to challenge how parties nominate candidates in 1981 and it went to the Supreme Court. The court decided that states can't interfere with how parties nominate candidates. Details are below:
https://www.oyez.org/cases/1980/79-1631
Question
Can a state compel a National Party Convention to admit delegates from its state if those delegates were selected in a manner violating the National Party's rules?
Ruling:
No. In a 6-3 opinion authored by Justice Potter Stewart the Court reversed the decision of the Wisconsin Supreme Court. Referring to its decision in Cousins v. Wigoda, the Court held that the "First Amendment freedom to gather in association for the purpose of advancing shared beliefs is protected by the Fourteenth Amendment from infringement by any State." Granting states the power to place delegates in a National Convention over party objections to how they were selected would impair the right of political parties to associate with whom they wish, especially when non-party members could influence the interests of the delegates. Justice Lewis Powell argued in his dissent that Wisconsin's "compelling interest" to involve non-affiliated voters in the primary justified this burden on the freedom of association, although the majority held that this could be accomplished without intruding upon internal party rules.
https://www.oyez.org/cases/1980/79-1631
This video contains a breakdown: