Is Sarima(0,0,0)(5,0,0,12) the same as the 5 year avg of a given month? Trying to understand big P,D,Q in sarima better. Could probably ask a broader question to get more info but my first question is making sure I understand at least the seasonal AR term.
With m=12 (monthly data), say the month being forecast is May, would Sarima(0,0,0)(5,0,0) just return the average of the value for May the past 5 years?

Your SARIMA model above should not be said to simply return the moving average of the (May monthly) values for the past 5 years, instead it has a seasonal autoregressive component via using the past 5 values of the time series at a frequency of 12 to make a forecast for the next value. Its specific forecast result would form a weighted linear combination of those 5 values and the weights are dependent on the characteristic of your sample time series dataset being fit to. Finally keep in mind in time series modeling the moving average process is conceptually different from moving average proper as emphasized in the reference:

The moving-average model should not be confused with the moving average, a distinct concept despite some similarities. Contrary to the AR model, the finite MA model is always stationary... The moving-average model is essentially a finite impulse response filter applied to white noise, with some additional interpretation placed on it... in the MA model a shock affects $X$ values only for the current period and $q$ periods into the future; in contrast, in the AR model a shock affects $X$ values infinitely far into the future

