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 existence of unique NE solution to the MFG, which is an $ε$-NE approximation to the $N$-player game, with $ε=O\left(\frac{1}{\sqrt{N}}\right)$. As an example, we analyze in details a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player's optimal strategy, with sensitivity analysis of the model parameters.