  Gauge fields coupled to dynamical matter are ubiquitous in many disciplines
of physics, ranging from particle to condensed matter physics, but their
implementation in large-scale quantum simulators remains challenging. Here we
propose a realistic scheme for Rydberg atom array experiments in which a
$\mathbb{Z}_2$ gauge structure with dynamical charges emerges on experimentally
relevant timescales from only local two-body interactions and one-body terms in
two spatial dimensions. The scheme enables the experimental study of a variety
of models, including $(2+1)$D $\mathbb{Z}_2$ lattice gauge theories coupled to
different types of dynamical matter and quantum dimer models on the honeycomb
lattice, for which we derive effective Hamiltonians. We discuss ground-state
phase diagrams of the experimentally most relevant effective $\mathbb{Z}_2$
lattice gauge theories with dynamical matter featuring various confined and
deconfined, quantum spin liquid phases. Further, we present selected probes
with immediate experimental relevance, including signatures of disorder-free
localization and a thermal deconfinement transition of two charges.
