Here’s a post that discusses in general how to get a posterior predictive distribution: How to get samples from the posterior predictive distribtion using stan
It seems to me that the missing piece of the puzzle is a function that can return a random sample of a Wald distribution. I’m not sure how you would do that within Stan itself, but you can use R, Python, or whatever language you prefer to post-process the MCMC samples that you’ve obtained from Stan in order to generate a posterior predictive distribution. It looks like there are functions available for both R and Python that can generate samples from a Wald distribution.