How essential to modern cosmology is the vacuum energy concept, and what would happen to our model of the expanding universe if it were proved to be non-existent?
I'm neither a quantum physicist, nor a cosmologist, so I'm going only from my laymans understanding.
The framework for constructing theories to explain everything in the universe except gravity which has proved most popular and useful for the last 50 or so years, is quantum field theory. In this theory, there are multiple fields (conceptually similar to the electric and magnetic fields) suffusing every point in space and time. From the values of these fields, we can, in principle, compute the probabilities of the outcomes of any experiment, including the probability of detecting a particular kind of particle at a particular place and time, and from the rules of these computations we can derive all the normal "laws of physics" such as the conservation of energy, the types of particles which exist and so on, as "emergent" properties.
I described QFT as a "framework" because there are many possible sets of fields and rules for their interactions, each of which gives different physics. From the experiments we can do, we can develop some constraints on the overall structure, but we also know that the fields and interactions that we know about cannot be the whole story. Over short distance and time scales and large energy scales they would give rise to non-sensical predictions, so they must be wrong, but we cannot yet do experiments to find out what really happens.
Anyway, the vacuum is the configuration of those fields which has the lowest possible energy, but that does not have all the fields being zero, and when we calculate that minimal energy (as far we can) using only the known particles and fields we get a ridiculously high value (that's one of the non-sensical predictions I mentioned). This is not taken as a sign that the vacuum energy is that high, but that the unknown high-energy fields must somehow contribute to lowering it.
None of this is very relevant to cosmology, which, except for the very earliest moments of the universe, does not really depend on the details of QFT, but only on the "emergent" properties such as the behaviour of gasses, stars, light waves, neutrinos, etc. and on the one thing that (so far) does not seem to fit into QFT, namely gravity (or general relativity, if you prefer). For, sure, if the vacuum energy actually were the silly value computing from known forces using QFT, the gravitational effects of that energy would make anything like the universe we see impossible, but that is just taken as confirming that there is as yet unexplored physics on small scales.
So, the summary is "not essential at all". Cosmology and the expanding universe depends on GR and properties of matter and radiation on large scales and at relatively low energies. The Vacuum energy relates to the properties of the fields that underly that matter and energy on very small scales.