  We consider random walks on the support of a random purely atomic measure on
$\mathbb{R}^d$ with random jump probability rates. The jump range can be
unbounded. The purely atomic measure is reversible for the random walk and
stationary for the action of the group $\mathbb{G}=\mathbb{R}^d$ or
$\mathbb{G}=\mathbb{Z}^d$. By combining two-scale convergence and Palm theory
for $\mathbb{G}$-stationary random measures and by developing a cut-off
procedure, under suitable second moment conditions we prove for almost all
environments the homogenization for the massive Poisson equation of the
associated Markov generators. In addition, we obtain the quenched convergence
of the $L^2$-Markov semigroup and resolvent of the diffusively rescaled random
walk to the corresponding ones of the Brownian motion with covariance matrix
$2D$. For symmetric jump rates, the above convergence plays a crucial role in
the derivation of hydrodynamic limits when considering multiple random walks
with site-exclusion or zero range interaction. We do not require any
ellipticity assumption, neither non-degeneracy of the homogenized matrix $D$.
Our results cover a large family of models, including e.g. random conductance
models on $\mathbb{Z}^d$ and on general lattices (possibly with long
conductances), Mott variable range hopping, simple random walks on Delaunay
triangulations, simple random walks on supercritical percolation clusters.
