A moderated regression model is an ordinary regression model with a product term for each moderator. For this model to be reasonable the residuals reassured to be sampled from a normal distribution of independent errors with constant variance (variance that is not a function of any predictors).
So yes the residuals need to be approximately normal for inferences in the model to be useful. As sample sizes increase this matters less and inferences will be reasonable accurate (assuming the central limit theorem applies), but it can require very large samples for this to be true if there is severe skew or kurtosis for instance.
thank you both! And what about the bootstrapping that appears to happen when carrying out the PROCESS moderation analysis? Doesn't this correct for nonnormality if that appears to be the case?