Future quantum computers will enable novel sign-problem-free studies of dynamical phenomena in non-perturbative quantum field theories, including real-time evolution and spontaneous supersymmetry breaking. We are investigating applications of quantum computing to low-dimensional supersymmetric lattice systems that can serve as testbeds for existing and near-future quantum devices. Here we present initial results for the $\mathcal{N} = 1$ Wess--Zumino model in 1+1~dimensions, building on our prior analyses of 0+1-dimensional supersymmetric quantum mechanics. In addition to exploring supersymmetry breaking using the variational quantum eigensolver, we consider the prospects for real-time evolution.