  The residual finite-dimensionality of a $\mathrm{C}^*$-algebra is known to be
encoded in a topological property of its space of representations, stating that
finite-dimensional representations should be dense therein. We extend this
paradigm to general (possibly non-self-adjoint) operator algebras. While
numerous subtleties emerge in this greater generality, we exhibit novel tools
for constructing finite-dimensional approximations. One such tool is a notion
of a residually finite-dimensional coaction of a semigroup on an operator
algebra, which allows us to construct finite-dimensional approximations for
operator algebras of functions and operator algebras of semigroups. Our
investigation is intimately related to the question of whether residual
finite-dimensionality of an operator algebra is inherited by its maximal
$\mathrm{C}^*$-cover, which we resolve in many cases of interest.
