So that suggests (and I recall other instances) that there can be legal action taken -- at least in states that have primaries run by the states, so those are not private elections -- against the parties, which are private entities.  

That is, your question confuses a lot of the ownership of who does what.  That suit was not about what the state did or didn't do, as the states do not apportion the delegates; states just hold the primary elections, and then the parties determine apportionment of delegates from the results of the state primary elections.

The vote at a party convention is, of course, a private election, but that's later in the timeframe -- and your question on whether a party could be legally "constrained" from nominating a candidate discovered to have engaged in vote fraud is moot, as the nomination would have occurred.  Nor did the agenda before the Dems' rules committee at the infamous meeting of May 2008 deal with vote fraud, as that is defined.  It was more, well, delegate suppression, as that term is used to delineate different ways in elections of messing with the will of the people.

Now, none of this deals wiht caucuses.  States do not run caucuses; state parties do.  I certainly can see grounds for suits against state parties for some caucus practices, such as violations of ADA, too, but I'm not recalling such suits.

Btw vs. an implication in a comment above, the May 2008 meeting of the national Dems' rules committee was sought by Obama's forces as much as those of other candidates -- at the time, it was said, because the outcome of the suit did not look good for Obama, and because the chair of the rules committee, Brazile, was among Obama's forces.  (Yes, the path was laid for years, with smart strategy such as that internal tactic.)

And that the result of the rules committee was wrong vs. the charge of the committee, the intent of the rules as passed by the party delegates at party conventions, etc., is a different question from whether the result was illegal.