Electromagnetic energy is released only by radiative transitions between molecular or atomic energy states (rotational, vibrational, electronic, etc.). When a molecule is promoted into a high energy vibrational state, for example, it may transfer to a lower energy state by emitting a photon with (approximately) the same energy as the difference in energy between the two states. In a perfectly isolated, vibrationally excited molecule, the likelihood of photon emission via this route is high, because it represents the only route to relaxation from the higher energy state to the lower energy state. But when the molecule is permitted to interact with other molecules, a collision may occur, which instead transfers the excited energy between the two molecules rather than emitting it as a photon. In this sense, conductive energy (heat) transfer competes directly with radiative energy transfer. The higher the concentration of molecules, the more frequent the collisions, the more likely conductive energy transfer outcompetes radiative energy transfer. In condensed phases, there is very little radiative emission observed because the efficiency of collisions is so high - excited molecular states just don't stick around long enough to emit photons. Bear in mind, this is all relative - there is ALWAYS a nonzero probability of radiative emission. But if the probability is low enough, you will not be able to detect it.
The amount of radiative transfer events is also proportionate to the number of molecular excited states at any one time, however. Therefore even if the probability of radiatve transfer is very low due to collisions, if you form enough of them at a given time, just by probability you can measure the radiative events. The most common way to increase the number/concentration of molecularly excited states in a lump of matter is to heat it up. As you feed thermal energy into a system, there is a statistical distribution of molecularly excited states. The more energy, the more excited states, the more radiative conversion you will observe. This is basically why very hot objects glow while they also give off heat.
Back to the case you bring up here - in principle there is both radiative transfer of energy and conductive transfer of energy (heat transfer) away from the point of reaction. If you had a sensitive enough instrument, you could probably pick the radiative component out above the background (which also emits thermal radiation at any temperature above absolute zero). But this process is carried out in condensed phase, and efficiency of heat conduction is very high such that energy produced by the reaction is not able to concentrate enough to generate a high proportion of molecular excited states. If you could increase the rate of energy flow and reaction events enough, conductive heat transfer wouldn't be able to spread out the energy fast enough, and the system would probably start to glow. But the amount of heat would be pretty substantial, so other parts of your system would likely fail.
(This is more or less what happens in an incandescent light bulb - you drive a current through a resistive wire fast enough, it heats up quickly and there's nowhere for the thermal energy to go before radiative transfer becomes efficient. This is especially the case in a light bulb, which is evacuated to reduce conduction of the resistive heating away from the filament - and prevent reaction with oxygen, of course, that would destroy the system in the process.)