Let $M$ be an acyclic NFA.
Since $M$ is acyclic, $L(M)$ is finite.
In a related question, it was suggested that exact counting of the number of words accepted by $M$ is $\#P$-Complete.
The second answer for that question provides a counting algorithm, but only works for unambiguous NFAs (where every word is accepted by at most a single path).
Given an NFA $M$, can we approximate $|L(M)|$ in polynomial time?
As automata is a highly studied subject, I was surprised that I couldn't find anything about this, so if someone knows of a reference it'll be great :).