Welcome!
You can check out the code associated with chapter 22 of Kruschke’s DBDA book. The code doesn’t reflect recent versions of pymc, but it makes a decent intro. There have also been a few discussions of multinomial regression models, for example, this one and this this one. If those don’t help or if you need help adapting the specifics of your model (e.g., the random intercepts), feel free to ask.