  Given a lattice polytope $P$ and a prime $p$, we define a function from the
set of primitive symplectic $p$-adic lattices to the rationals that extracts
the $\ell$th coefficient of the Ehrhart polynomial of $P$ relative to the given
lattice. Inspired by work of Gunnells and Rodriguez-Villegas in type
$\mathsf{A}$, we show that these functions are eigenfunctions of a suitably
defined action of the spherical symplectic Hecke algebra. Although they depend
significantly on the polytope $P$, their eigenvalues are independent of $P$ and
expressed as polynomials in $p$. We define local zeta functions that enumerate
the values of these Hecke eigenfunctions on the vertices of the affine
Bruhat--Tits buildings associated with $p$-adic symplectic groups. We compute
these zeta functions by enumerating $p$-adic lattices by their elementary
divisors and, simultaneously, one Hermite parameter. We report on a general
functional equation satisfied by these local zeta functions, confirming a
conjecture of Vankov.
