Why do photons within one body have different energies in a thermal radiation spectrum? Does it have to do with different modes of vibrations or phonons (for solid states)? In other words, is the temperature of a body's same atoms at all points equal which means they must emit the same photons with the same energy?
The radiating photons which are emitted from the surface of a hot body have a continuous distribution of different energies (which form the blackbody spectrum) because the electrons which belong to the atoms whose vibrations cause the emission experience a distribution of accelerations. Then that distributions gets shaped smooth by the continuous exchange of energy between the photons and the electrons and reaches its final form when the radiation and the surface come into thermal equilibrium.
The details of the high-energy and low-energy cutoffs on the spectrum are complicated and best studied by searching on blackbody spectrum.