Does the re-arrangement of chemical bonds happens due to electromagnetic interactions? The question is of some interest because the storage of energy in a recharable battery is not caused by gravitational (potential) nor by kinetic energy.
From this question  Why is current the same in a series circuit? it is clear that the release of energy in a circuit is due to energy loss in the form of EM radiation of each involved electron. The answers are stating that the number of the electrons is unchanged and the charge is unchange too of course. 
So the question is, does the re-arrangement of chemical bonds happens due to electromagnetic interactions? And is some part of the EM radiation stored inside the bonds and will be released during the flow through Ohm resistances?
 A: In a rechargeable battery, two types of reversible chemical reactions take place:


*

*Oxidation reaction: in which a chemical, referred to as the reducing agent ($Re$) is oxidised by donating electrons:


$$Re \to Re^{z+} + z e^-$$


*Reduction reaction: in which a chemical, referred to as the oxidising agent ($Ox$) is reduced by receiving electrons:


$$Ox + z e^- \to Ox^{z-}$$
The overall reaction is thus:
$$Re + Ox \to Re^{z+} + Ox^{z-}$$
The reaction is chosen so that the reaction equilibrium constant:
$$\frac{a_{Rez+}{a_{Oxz-}}}{a_{Re}a_{Ox}}=K$$
... is much larger than unity, so that the reaction is thermodynamically favourable, from left to right.
The battery is made of cells (arranged in series or parallel, or a single cell in some cases), each running such a reaction and providing a flow of electrons when the circuit is closed.
This provision of current continues until all the left hand reagents have been converted to right hand reaction products. Then the battery can no longer provide current and the potential across its electrodes is zero.
But remember that the chosen reactions are reversible: by running a current through a depleted battery and in the reverse direction of discharging mode, the reagents are converted back to the original reagents and the battery is as new.
Note that even with non-rechargeable batteries some degree of recharging is in fact possible but they happen to be much harder to recharge than those designed for that purpose.
The chemical energy stored in chemical compounds is due to changes in the Molecular Orbital structures when reagents convert to reaction products. Take a simple combustion reaction (the burning of coal):
$$C(s) + O_2(g) \to CO_2(g)$$
Once initiated, such a reaction proceeds spontaneously because the overall arrangement of Molecular Orbitals is of a lower level of energy for $CO_2(g)$ than for $C(s) + O_2(g)$. Energy is thus released, here mainly as heat.
In the case of a battery, this reaction energy is released as EMF due to the particular reactions chosen and the physical arrangement of the cell(s).
