As far as I know not from direct measurement like you can get band gap or work function. You can get comparative information such as which end of the molecule is more polar, or similar information. This would be done by knowing something about the surface.

Edit:

I have spent the last couple of days thinking about this and I think there may be a way to measure the dipole. If you assume the molecule is perfectly flat, lies parallel to the surface, and operated the STM in constant height and constant current (only voltage changes), then any change in voltage would be due to changes in electron density of the molecule. You could then in essence take the derivative of the change in voltage along the molecule to determine the dipole.

The problem is that the matrix interaction term in STM tunneling would also change based on the element the tip interacts with. So all your results that have different elements would give results that you would expect from looking at the periodic table, where oxygen would be brighter than carbon and so forth.

I would suggest using spectroscopy with linearly polarized radiation. If the molecule is adsorbed on the surface of a solid, then turn the polarization plane relative to the adsorption layer. Then, probably, it will be possible to observe a difference in absorption when the plane of polarization is oriented along and across the dipole axis.

Since the absorption is associated with reflection and transmission, then on these spectra we can also expect features corresponding to the orientation of the plane of polarization along and across the dipole axis.