  We establish two theorems for assessing the accuracy in total variation of
multivariate discrete normal approximation to the distribution of an integer
valued random vector $W$. The first is for sums of random vectors whose
dependence structure is local. The second applies to random vectors~$W$
resulting from integrating the $\mathbb{Z}^d$-valued marks of a marked point
process with respect to its ground process. The error bounds are of magnitude
comparable to those given in Rinott and Rotar (1996), but now with respect to
the stronger total variation distance. Instead of requiring the summands to be
bounded, we make third moment assumptions. We demonstrate the use of the
theorems in four applications: monochrome edges in vertex coloured graphs,
induced triangles and $2$-stars in random geometric graphs, the times spent in
different states by an irreducible and aperiodic finite Markov chain, and the
maximal points in different regions of a homogeneous Poisson point process.
