There are two ways to state this question, and the answers differ.

1) Let the dimension $d$ is fixed, and the input is via $n$ vertices of the polytopes. In this case the total number of possible simplices is polynomial, and so is the counting problem.

2) If the dimension $d$ is arbitrary, everything falls apart. Think of a simplex with vertices $O=(0,\ldots,0)$ and $(0,\ldots,a,\ldots,0)$, where $a\in$ {$a_1,\ldots,a_m$}. Here $n=dm+1$. Let $z=(c,\ldots,c)$ and consider all (closed) simplices which contain $z$. Check that #simplices containing O is a variation on the #knapsack problem, several versions of which are known to be #P-complete. I omit the details.