I know that an iron coil channels and concentrates the magnetic flux of a coil inductor, but does an iron core increase the total energy capacity of the coil at peak current? The iron increases the inductance which should increase the energy capacity right? The energy takes longer to induce into the coil, but at the same peak current the inductor has now stored more energy right?
Yes, an iron core increases the inductance of a coil, and at any given exciting current, that also increases the stored energy.
Ferromagnetic materials (iron being the most familiar) do not merely channel the field, but take on polarization (which is effectively extra circulating current that does not go through the external electric circuit). That effect is not unlimited, because only a few electrons are free to take new spin and orbit alignment. Once you get to 100% alignment, the magnetic polarization response to additional current is zero.
This effect, 'saturation', sets a maximum current, above which a coil's inductance is no longer enhanced by the core. That complicates the claim that "at the same peak current the inductor has now stored more energy". Peaks of current might be excursions into saturation.