We consider a general class of nonzero-sum $N$-player stochastic games with
impulse controls, where players control the underlying dynamics with discrete
interventions. We adopt a verification approach and provide sufficient
conditions for the Nash equilibria (NEs) of the game. We then consider the
limit situation of $N \to \infty$, that is, a suitable mean-field game (MFG)
with impulse controls. We show that under appropriate technical conditions, the
MFG is an $\epsilon$-NE approximation to the $N$-player game, with
$\epsilon=\frac{1}{\sqrt{N}}$. As an example, we analyze in details a class of
stochastic games which extends the classical cash management problem to the
game setting. In particular, we characterize the NEs for its two-player case
and compare the results to the single-player case, showing the impact of
competition on the player's optimal strategy, with sensitivity analysis of the
model parameters.