Yes, $n!\bmod p$ is computable in FCH. More generally, if $f$ is a polynomial-time computable function, then given $n$ and $m$ in binary, we can compute $$\prod_{i<n}f(i)\bmod m$$ in FCH. This follows from the fact that if we are given in unary $n$, $m$, and a sequence of numbers $a_0,\dots,a_{n-1}$, then we can compute $$\prod_{i<n}a_i\bmod m$$ in uniform $\mathrm{TC}^0$, which was proved by
William Hesse, Eric Allender, and David A. Mix Barrington: Uniform constant-depth threshold circuits for division and iterated multiplication, Journal of Computer and System Sciences 65 (2002), no. 4, pp. 695–716, doi 10.1016/S0022-0000(02)00025-9.