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In this question there is a value given for absorptivity of infrared radiation however in order to work out a value for heat flux emitted from the solar panel in order to answer part (a) surely I'd need a value of emissivity? I have been stuck on this and have no clue how to work part (a) out without a value for emissivity.
You don't get the the emissivity coefficient because it is already factored in the problem. You have a constant temperature of $71[^oC]$ on the external surface, and this is explicitly stated as a steady state problem.
So you only need to consider other types of heat transfer (e.g. conductive).
