Do objects have energy because of their charge? My gut feeling tells me things should have energy because of their charge, like they have energy because of their mass.
Is this possible? Has it been shown? If not then what is missing to make such an equivalence possible?
 A: 
My gut feeling tells me things should have energy because of their charge, like they have energy because of their mass.

You are not the first one who had this idea. There was indeed a concept calles "electromagnetic mass", where the electrostatic energy $E_{em}$ of a charged particle at rest would be
$$E_{em}=\frac{e^{2}}{2\cdot r}$$
(where $e$ is the charge and $r$ the radius of the particle)
and because energy and mass are equivalent the electrostatic mass $M_{em}$ should be
$$M_{em}=\frac{2\cdot e^{2}}{3\cdot r\cdot c^{2}}$$
To quote from Wikipedia:
Wilhelm Wien and Max Abraham came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass.
Of course also the weak and strong force would contribute as well as electromagnetism.
A: The problem with your question, and the reason you have so many comments asking for clarification, is that energy is a slippery concept. Generally speaking we are interested in energy differences. So, for example, if you consider a two charged particles it's easy to calculate the energy change as you bring them together. By contrast, if you have a universe with just  single electron in it, it's not at all clear what you mean by the energy of the electron. One of the comments referred to the electron self energy, but classically this is infinite. Even if you consider quantum mechanics the self energy is infinite until you turn it into a difference.
But let me suggest a way of looking at it that you might find interesting. NB this isn't an answer, because I'm not sure your question has an answer as it stands, but it is one perspective.
Although we normally consider energy differences, we consider mass to be absolute. After all, a body can be massless or have a finite rest mass, and this is generally unambiguous. But we know that energy and mass are related by Einstein's famous equation $E = mc^2$, so if the charge on an electron increases its energy it must also increase its mass. Mass comes in two flavours: inertial mass and gravitational mass (Einstein tells us these are the same thing). We can't do much with the inertial mass because we don't have an uncharged electron to compare to a charged electron, but we can look at the gravitational mass.
The gravitational field of an isolated, spherically symmetric, charged object (like an electron) is given by the Reissner-Nordström metric. This is somewhat opaque for the non-nerd, but let's ask a simple question: how does the escape velocity for the charged body depend on the charge?
The escape velocity is given by:
$$ v = -\sqrt{\frac{2G}{r} \left( M - \frac{Q^2}{2r} \right)} $$
where $M$ is the mass of the object and $Q$ is its charge. However this tells us something rather strange. As you increase the charge the escape velocity decreases, and in fact if you increase the charge enough the escape velocity falls to zero. So a charged body has a lower gravity than an uncharged body of identical mass.
Now it almost certainly makes no sense to describe an electron as a Reissner-Nordström black hole. Apart from anything else its event horizon would be many orders of magnitude smaller than the Planck length and you'd expect some so far unknown theory of quantum gravity to take over from General Relativity and change its predictions. Nevertheless, you could use the above reasoning to claim that a charged electron actually has a lower energy than an uncharged one would. Now there's an unexpected result :-)
A: I don't know if this will answer your question but here goes...
Electric charges have an electrostatic potential energy which is the amount of work done in moving a charge in an electric field against forces of attraction or repulsion. If this electric field is produced by an unit charge, then it's called $\text{voltage}$.
When moving charges the constant $\frac{1}{4\pi \epsilon}$ determines the amount of work done to assemble the charges. For an isolated charge, you can't determine the potential energy unless you bring a test charge into the field.
Energy can't be obtained from stationary charges and there's no rest energy for a mass-less but charged particle.  Keep in mind, energy can be obtained from mass irrespective of its motion ($E = mc^2$). The same is true of momentum, kinetic energy, etc.  These depend on mass only and not charge, magnetism, etc.
