As you can see, it’s very hard to answer these questions in general. Have you looked through the Stan manual to see what it can express? Check out the latent discrete parameters chapter (and missing data chapter if you have missingness). Sometimes the missingness doesn’t even matter, so it will depend on the model.
Are you familiar with BUGS, which makes this translation very easy? (But then it can be very challenging to fit with lots of discrete parameters or if there is high posterior correlation among parameters.)
As @bgoodri pointed out, there’s no way to implement discrete parameters in Stan directly—they have to be marginalized out if you want to work in Stan. The manual chapters I cited show how to do that and why it’s much more effective than sampling them (even if you could do pure MC samples of the discrete parameters).