The ARMA equation is just a recursive equation for the time series, but it is usually paired with an assumption that the error terms in the model are white noise (i.e., IID normal with zero mean). Moreover, if you want the model to be strongly stationary the resulting joint distribution of the time-series values should be normal. (You can find more on the latter aspect of the process in related questions here and here). So as a practical matter, while it is possible to form ARMA models that are not normal, these are also non-stationary and so they are not in common use.
In any case, there is no stationary ARMA process with a non-empty auto-regressive part that can model a count variable (i.e., a non-negative integer). The reason for this is that any stationary non-empty AR-part will always have a coefficient that is not an integer, so it will multiply previous integer count values by a non-integer, which will give a non-integer output in at least some cases. (As an exercise, you might want to have a look at the ARMA model form and see if you can demonstrate this.)
As jbowman points out in the comments, what you can do is to have a hierarchical model where you specify a distribution for the observable count values and then model some parameter of this distribution (e.g., the logarithm of the mean of the count) as a latent variable using an underlying ARMA model. This type of model would allow you to have count values as your observable variable while still exhibiting latent auto-regressive behaviour.