Insofar as the blackbody spectrum is a "reasonably" good approximation for the radiation of the sun and presumably every ordinary star, and discounting objects which merely reflect light of some colour, are there any notable astronomical objects which emit radiation but are not well-approximated as blackbodies?
yes, there are, as follows. I'll list the ones I know and invite others here to add whatever I leave out.
First of all, there are synchrotron sources which comprise fast-moving electron beams interacting with very strong magnetic fields, causing the electrons to orbit the axis of the beam and radiate electromagnetic waves at radio frequencies. their spectrum is distinctly non-thermal, following some form of power law.
Then there is the so-called 21-centimeter line, which is microwave radiation corresponding to an energy transition in neutral intergalactic hydrogen. It is used to map the distribution of hydrogen in space. Redshifts for these clouds allow astronomers to infer their distances and hence their ages.
Then there are sources associated with particle/antiparticle annhilation, often of energy ~0.511MeV (for positron/electron processes) which might take place in the vicinity of supermassive objects.