Do Vector Autoregression models have the same p, and q order terms as a ARMA model or same number of ACF/PACF? Do Vector Autoregression models have the same p, and q order terms as a ARMA model?
Do you have n (Partial) autocorrelation function plots (P)ACF, one for each of the n time-series, or do you still have one ACF and PACF plot?
 A: Lag orders
VAR does not have any moving average ($q$) order, as it does not have a moving average component. VAR is a multivariate counterpart of AR, not of ARMA. (There is the VARMA model which does have the $q$ order, but the model is considerably less popular than VAR.)
Also, the autoregressive ($p$) order of a VAR model does not have to coincide with the $p$ orders of the corresponding AR models of the univariate series. Since an equation of $y_{i,t}$ in a VAR model includes both own lags $y_{i,t-k}$ and cross-lags $y_{j,t-\ell}$, there is no direct correspondence to the AR model for $y_{i,t}$ that only includes the dependent variable's own lags $y_{i,t-k}$.

ACF, PACF and CCF
VAR is a model for a multivariate time series. A single ACF or PACF plot characterizes a univariate time series. In the multivariate case, every time series has its own ACF and PACF plots. In addition, you can examine pairwise cross correlation functions (CCF).
In R, you can get both ACF and CCF by applying the acf function on a multivariate time series object. Try e.g.
library(vars)
acf(Canada)

