We establish a quantitative normal approximation result for sums of random variables with multilevel local dependencies. As a corollary, we obtain a quantitative normal approximation result for linear functionals of random fields which may be approximated well by random fields with finite dependency range. Such random fields occur for example in the homogenization of linear elliptic equations with random coefficient fields. In particular, our result allows for the derivation of a quantitative normal approximation result for the approximation of effective coefficients in stochastic homogenization in the setting of coefficient fields with finite range of dependence. The proof of our normal approximation theorem is based on a suitable adaption of Stein's method and requires a different estimation strategy for terms originating from long-range dependencies as opposed to terms stemming from short-range dependencies.