  Quantitative multivariate central limit theorems for general functionals of
possibly non-symmetric and non-homogeneous infinite Rademacher sequences are
proved by combining discrete Malliavin calculus with the smart path method for
normal approximation. In particular, a discrete multivariate second-order
Poincar\'e inequality is developed. As a first application, the normal
approximation of vectors of subgraph counting statistics in the
Erd\H{o}s-R\'enyi random graph is considered. In this context, we further
specialize to the normal approximation of vectors of vertex degrees. In a
second application we prove a quantitative multivariate central limit theorem
for vectors of intrinsic volumes induced by random cubical complexes.
