By conservation of energy, the solid is left in a lower energy state following emission of a photon. Clearly absorption and emission balance at thermal equilibrium, however, thermodynamic equilibrium is a statement of the mean behaviour of the system, not a statement that the internal energy is constant on arbitrarily short timescales. The energy has to come from somewhere during emission, and go somewhere during absorption.
Energy in a solid can be stored as kinetic and potential energy of electrons and nuclei, either individually or in collective modes such as phonons and plasmons. In thermal equilibrium energy will be stored more or less in various forms depending on the temperature and material. However, even if most of the thermal energy in a particular solid at temperature $T$ is stored in the form of phonons, it could be that phonons primarily interact with light indirectly via electrons, e.g. a phonon excites an electron in a phonon-electron interaction, which can interact with light via the EM field.
Given that light is an EM field, it makes sense to me that it is emitted and absorbed by charged particles. The electron-photon interaction is probably dominant for visible and ultraviolet light, given that metals are opaque, while semiconductors and insulators are transparent to (visible and UV) light with energy lower than their bandgap. However, once you get into energies in the IR and below, or X-rays and above, other mechanisms apparently take over. For example, on the high-energy end of the spectrum I've heard that gamma rays can interact directly with nuclear degrees of freedom, which is reasonable considering that gamma rays are emitted during a lot of nuclear reactions.
A review of absorption spectroscopy might give clues to important light-matter interactions over a broad range of wavelengths. Whether all of the these processes are involved in blackbody emission is a somewhat different question.
What physical processes mediate energy transfer during blackbody emission, and in which energy ranges are the various processes dominant?
Best Answer
This is a fantastic question, and a topic I was very confused about when I first took a class on Radiative Processes. The ultimate answer, as hinted at by @LubošMotl, is anything---if you start with a 'white-noise' of radiation (i.e. equal amounts of every frequency), it will equilibrate with the medium/material into a black-body distribution because of its thermal properties (see: Kirchhoff's Law, and the Einstein Coefficients). This is just like if you gave each molecule in a gas the same energy, they would settle to a Boltzmann Distribution.
In practice (and hopefully a more satisfying answer) is that it's generally a combination of line-emission and Bremsstrahlung, with Bremsstrahlung1 dominating at high temperatures ( $T \gtrsim 10^6 -10^7 K$ ). Lines are produced at myriads of frequencies depending on the substance of interest, and the thermodynamic properties (e.g. temperature). For everyday objects, I think the emission is primarily from molecular-vibrational lines. Individual lines are spread out by numerous thermodynamic broadening effects to cover larger portions of the spectrum. Finally, as per Kirchhoff's law, equilibrated objects can only emit up to the black-body spectrum. In practice, you'll still see emission/absorption lines imprinted, and additional sources of radiation.
Lets look at a breakdown of the relevant transitions as a function of energy level:
radio: nuclear magnetic energy levels (also cyclotron emission in the presence of moderate magnetic fields).
microwave: rotational energy levels
infrared: vibrational energy levels (molecules)
visible: electronic (especially outer electron transitions)
ultraviolet: electronic (especially outer/valence electron ejection/combination)
x-ray: electronic (inner electron transitions)
gamma-ray: nuclear transitions
1: Bremsstrahlung (German for 'braking radiation') is radiation produced by the acceleration of charged particles---most often electrons. This can happen between any combination of bound (in atoms) or unbound (free or in plasma) charges.