I'm running a study with a between-groups component and a within-groups component, with a predictor that is count data. I was planning on using split-plot ANOVA to analyze this, but after seeing lots of questions on here, it is commonly said that ANOVAs are not good with count data, but none of the answers specified why. Is it simply an issue of count data usually not being normally distributed? If that is the case, would a split-plot ANOVA be appropriate if the count data were, in fact, normally distributed?
Count data cannot really have a normal distribution, except as an approximation in the case of large counts. But that is not the main reason:
Count data do not have constant variance. For the most used model, the Poisson distribution, the variance equals the mean, for many other reasonable models (cluster processes, like people arriving at some place in groups), variance is proportional to the mean. A classical solution which is sometimes useful even today is the square root transformation, stabilizing variances. See Why is the square root transformation recommended for count data?.
Counts are extensive variables, not intensive. See Goodness of fit and which model to choose linear regression or Poisson
As mentioned in a comment, ordinal regression models is a possibility. See GLM with continuous data piled up at zero for details, especially the answer by F. Harrell.
Today, with count data mostly think poisson-regression (or a variant), which is a glm (generalized linear model), at least as a starting point. But you can still use anova thinking, as all experimental design ideas used in anova can also be used with generalized linear models! So you can still use a split-plot model with poisson regression.