This news story has more information about the Arizona Democratic Party lawsuit to remove No Labels Party from the ballot than any other I have seen. However, the story implies that the lawsuit involves federal campaign laws. Although the Democratic Party Complaint mentioned that, the Complaint does not ask the judge to remove No Labels on the basis of campaign laws. The sole request is that the party be removed from the ballot on the grounds that its officers filed some paperwork a few weeks too early.
Also, the story says that if No Labels is on the ballot, it may or may not run any candidates. However, assuming it gets on the Arizona ballot, the leaders of No Labels have no authority to prevent Arizona voters from filing to run for office in the No Labels Party primary, for congress and partisan state office. In Arizona, all qualified parties nominate by primary.
The lawsuit is moving slowly. According to the website of the Maricopa County Superior Court, no attorney has filed yet as the representative of No Labels.