What is the source of vacuum energy during the inflationary era? It is assumed that during the inflationary epoch the universe is dominated by the vacuum energy. What is the source of this vacuum energy and how it that related to the energy of the inflaton field?  
 A: The general idea is that during the inflationary epoch there was a scalar field permeating through space that had a very large potential energy density. Configurations of field (like the electromagnetic field) can have both kinetic and potential energy associated with them. 
If such a field had a very high potential energy density associated with it, for example, because the field had a nonzero (vacuum expectation) value everywhere in space that would have a lot of potential energy associated with it, most of the energy of the field would be carried by its potential energy, and not by its kinetic energy density. Since the energy density of a field is proportional to 
$$
T^\mu_{\;\;\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\nu \phi - \mathcal{L}\delta^\mu_{\;\;\nu}
$$
where $\mathcal{L} \sim P - V $ is the Lagrangian density of the field (which is proportional to its potential energy density minus its kinetic energy density), we see that if the potential energy density of the field is much larger than the field's kinetic energy density, the resulting energy-momentum tensor will behave like a negative energy density.
In most models, people tune the dynamics of the inflation field in such a way that as the universe rapidly expands, this field slowly loses its potential energy and the expansion process comes to a halt. There is no precise relation between the amount of potential energy stored in the inflation field at the beginning of the expansion process and some sort of energy that will be stored in the spacetime afterward. Part of the reason is that in an expanding universe we can no longer define a notion of 'globally conserved energy', but that is another topic. 
How the inflationary field got into this particular configuration and how this potential energy got 'stored' in the inflation field in the first place, is usually not addressed. We don't know that happened before the inflationary epoch or how the universe behaves at even higher energies, the main focus is on trying to derive a model (withing the modern framework of (quantum) field theory) that answers several questions in cosmology (like the horizon and flatness problems) and give at the same time an accurate prediction of e.g. the distribution of hot and cold spots on the CMB background. Inflation is a general framework that describes how this could have happened using field theory, but of course, we can always ask 'what happened before that' or 'where did this field come from'. Perhaps in the future, it will turn out that the 'inflation field' is only an effective description of an even bigger overarching theoretical framework? Untill then we don't know and people try to come up with the simplest model that best fits the available data.
A: I have addressed your question in three parts:

It is assumed that during the inflationary epoch the universe is dominated by the vacuum energy. 

During the inflationary epoch, the expansion of the universe was dominated by the inflaton field, which is different from the vacuum energy which currently dominates expansion. 

how it that related to the energy of the inflaton field?

Both vacuum energy and and the inflaton field dominated universes give rise to accelerating expansion but have important differences (see Quintessence). For a over-simplified rudimentary intuition about the energy conservation aspect, one can say that the field loses it's energy which is then used by the universe to expand but it must be kept in mind that this is an oversimplification.

What is the source of this vacuum energy 

According to Wikipedia:

Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. 

the reason why the vacuum has a non-zero energy contrary to what one expects can be basically attributed to the Heisenberg Uncertainty principle which forbids us from knowing the exact state of any field.  
For a rudimentary intuition about the energy conservation aspect, one can say that the field loses it's energy which is then used by the universe to expand but it must be kept in mind that this is quite an oversimplification of the complex underlying theory which is still not yet fully understood .
As for the origins of the inflaton field and why things were in the state they were, as JgL pointed out in his excellent in-depth answer, that is still an open question. 
