  We analyse three interacting vacuum dark energy models with the aim of
exploring whether the $H_0$ and $\sigma_8$ tensions can be simultaneously
resolved in such models. We present the first ever derivation of the covariant
gauge-invariant perturbation formalism for the interacting vacuum scenario,
and, for the sub-class of geodesic cold dark matter models, connect the
evolution of perturbation variables in this approach to the familiar
cosmological observables. We show how $H_0$ and $\sigma_8$ evolve in three
interacting vacuum models: firstly, a simple linear coupling between the vacuum
and cold dark matter; secondly, a coupling which mimics the behaviour of a
Chaplygin gas; and finally a coupling which mimics the Shan--Chen fluid dark
energy model. We identify, if any, the regions of parameter space which would
correspond to a simultaneous resolution of both tensions in these models. When
constraints from observational data are added, we show how all the models
described are constrained to be close to their $\Lambda$CDM limits.
