The hypothesis implies  that $M_k=[e^{\sum_{n=1}^k X_n}]$ is a supermartingale, with $M_0=1$. Then the optional stopping theorem for positive supermartingales implies the requested inequality. (see e.g. Williams' book "Probability with martingales"). 
Note that no moment conditions on the stopping time $\tau$ are needed,
just that it is an almost surely finite stopping time. Alternatively, look up "Wald's third identity"and apply it to the independent variables $e^{X_n}/[\mathbb{E}e^{X_n}]$ and the given stopping time. Note that the case of a general stopping time follows from the case of a bounded stopping time via Fatou's lemma.