In the United States, both primary elections and general elections are government-organized events. These elections are primarily established and regulated by statutes passed by state legislatures.

Depending on state laws, political parties may have the option of choosing not to participate in a state primary election. ("Caucuses" are a well-known alternative used by the parties of some states. But sometimes, a state party may even just decide to have neither a primary or caucus; for discussion of this in the context of the 2020 election cycle, see "[Republicans to scrap primaries and caucuses as Trump challengers cry foul][1]", *Politico*, Alex Isenstadt, 09/06/2019. However, it seems some states, such as Michigan, may have election laws that do not allow parties to opt out of participating in a state primary: see CDJB♦'s answer to https://politics.stackexchange.com/questions/71908/) 

In fact, it seems the Colorado Republican party has expressed intent to cancel its primary for the 2024 election if Trump is not on the ballot, and have a caucus instead, but it is apparently disputed whether the party is legally allowed to withdraw from the primary process at this point: the Colorado Department of State made a statement that withdrawal is not allowed by Colorado law.  ("[Colorado GOP threatens to shift to caucus system over Trump ruling](https://thehill.com/homenews/campaign/4369749-colorado-gop-threatens-to-shift-to-caucus-system-over-trump-ruling/#:~:text=“You%20won%27t%20have%20to,this%20is%20allowed%20to%20stand.”)", *The Hill,* SARAH FORTINSKY - 12/20/23 12:20)

When a party has a primary election in a state, it is subject to all Constitutionally valid state laws regulating the election.

Parties have First Amendment association rights that election laws must not infringe, so parties do get a say in certain things like whether non-party members can vote in the party's primary election ([California Democratic Party v. Jones][3], 2000). However, there are various other constitutional issues that can come into play, so First Amendment association rights are not a trump card in this context.

I wrote a longer post with more details as an answer to a broader past question here: [How do parties really choose candidates in the US?](https://politics.stackexchange.com/a/80635/7580)


  [1]: https://www.politico.com/story/2019/09/06/republicans-cancel-primaries-trump-challengers-1483126
  [2]: https://thehornnews.com/report-gop-to-cancel-primary-election-after-trumps-removal/
  [3]: https://en.wikipedia.org/wiki/California_Democratic_Party_v._Jones