The electromagnetic radiation comes from accelerated charged particles, mainly electrons and positrons.
The surface of a neutron star is not made of neutrons. It is a totally ionised gas of nuclei plus electrons, with a composition that could range from iron-peak elements to hydrogen and helium accreted from the interstellar medium.
As a result of their rotation and strong dipolar magnetic fields then there are also very strong electric fields at the surface of the neutron star. These fields rip electrons and protons from the surface and accelerate them away - electrons from the magnetic poles and protons at lower magnetic latitudes. The classic work on this is by Julian & Goldreich (1969).
The acceleration due to these electric fields is easily sufficient to overcome gravity. The particles are forced to spiral along the magnetic field lines by the Lorentz force, and produce synchrotron or curvature radiation directly. The photons from these processes are then capable of creating secondary electron-positron pairs that are accelerated in opposite directions, releasing more photons in a cascade.
No neutrons are harmed in the making of this radiation, other than the ones in nuclei at the surface of the neutron star that get bombarded by high energy accelerated electrons and positrons.