Which single organic molecule has strong absorbance throughout the visible range?
Quite a few dyestuffs will have strong absorption across the visible range of wavelengths, though few will have an entirely even absorbance. This won't matter much is they are concentrated enough and absorb most of the light at each wavelength.

Acid Black 1, for example, shows strong absorbance from the UV to just under 700 nm, which is most of the visible. Absorbance is stronger in the red but overall absorbance is large enough that it is sometimes used in black inks. See this paper for a sample absorbance spectrum:



There are quite a few other dyes with strong absorbance at most visible wavelengths and whether they appear black depends on the concentration.