The full claim is as follows:
At a kiddie beauty pageant, Haley and her sister were ushered off the stage because they were weren't black or white − fitting neither of the local prize categories.
This requires journalistic research but it's not implausible on its face.
Before 1968, Miss America did not accept black applicants, although Native Americans and Asians were allowed to participate by the 1950s.1 In early 20th century Miss America competitions, during the musical part of the pageant, black women appeared as slaves in the back of the stage while the white contestants sang plantation songs.2
In response to this, two ethnically segregated pageant systems arose in the 1950s: Miss Indian America and Miss Chinatown USA.3 Black women did not have a national system; beauty pageants were generally associated with segregated high schools and HBCUs.4
The first black contestant in Miss America appeared in 1970, and the first winner was not until 1983. It's at least plausible that during this transitional period, local child beauty pageants may have still been perceived as places where beauty was evaluated racially. However, that's far from a proof that this incident happened the way Haley recalls it.
1 Valerie Felita Kinloch, The Rhetoric of Black Bodies (2004)
2 The Most Beautiful Girl in the World:
Beauty Pageants and National Identity (1999)
3 Beauty, Bodies, and Boundaries: Pageants, Race and U.S. National Identity (2010)
4 Black Beauties: African American Pageant Queens in the Segregated South (2020)
Edit: I've been asked to add a source on the idea that Haley would have confused Southerners in the 1970s by being neither white nor black. This common reaction is well-represented in period fiction, but I'm not sure what non-fiction citation to use as evidence. Until someone suggests one, I recommend the film Mississippi Masala (1991).