As far as I know, in any primary election with which I have ever had contact, there are rules for getting on the ballot. You file a notice of candidacy. You circulate nominating petitions and gather signatures. If you get enough signatures -- provided these stand up to challenge if anyone gets pushy -- you are on the ballot. You may be a nuisance candidate. You may be a disruptor working on behalf of the other party. You may be an angry dissident. You may be the biggest fool on God's green earth. Heck, you might even be a black man trying to run as a Democrat in Tennessee in 1960, with the intent of breaking the racist stranglehold on the party. Dot the i's and cross the t's, and you have a right to run.
Party organizations will often officially endorse incumbents, but that shouldn't affect how a name appears on the ballot.
I have never heard of a party committee denying ballot access to someone who had filed properly, submitted enough signatures, and otherwise dotted the i's and crossed the t's.
Looks like it. There are two links above. This is from Fox 13 -
Tennessee Democratic Party removes state rep from ballot
MEMPHIS, Tenn. - (AP) - The Tennessee Democratic Party voted to remove a state representative from the party's primary ballot over allegations that he had Republican donor financing and voted contrary to the party.
https://www.fox13memphis.com/news/tennessee/tennessee-democratic/7LREEYT3GTJAW6AN23ATHQDVIU/