If we are talking about static or quasistatic field, then energy stored in a field is an energy which can be released by through a field source if that field is changed. So the energy could be attributed not to the field, but to sources of the field. For example, the energy stored in a superconducting coil may be released an transformed into heat by terminating the coil on a resistor. The same way the energy of the electric field may be attributed to the potential energy of the charges, so the change in the field energy will lead to a change in the kinetic (or other) energy of that charges.
However if we are dealing with electromagnetic radiation, then we should to consider the energy of the electromagnetic field as a separate entity. On a microscopic level it is carried out by photons. As to the macroscopic approach, scientists spent a lot of time with the ether because it was difficult to comprehend how energy could be transfered without medium. So , eventually they said that EM radiation is a materia itself.