Eddington waited for a total solar eclipse to happen to be able to observe gravitational lensing of the stars behind the Sun. And nowadays, amateurs can do the same thing.
Of course, the Moon is much less massive than the Sun, so it doesn't bend light as much. But since Eddington's time, our observation capabilities have greatly increased. Is it possible to detect the gravitational lensing of stars that are behind the Moon?
Measuring the gravitational deflection of light by the Moon is just out of reach of current observational techniques.
The angular deflection caused by the lensing of a distant background object by a foreground (nearby) object is given by $$\theta \simeq 4 \frac{GM}{Rc^2},$$ where $M$ is the mass of the lensing object and $R$ is the closest projected distance of the ray from the centre of the mass.
For the Moon, the maximum deflection would occur when the ray just grazes the limb of the Moon and would equal 26 micro-arcseconds.
At present the most accurate instruments for measuring precision positions are VLBI radio observations of point sources, where 10 micro-arcsecond relative positional precision can be "routinely achieved" (Reid & Honma 2014). To use this technique you need bright, radio point sources to be close in position to the Moon. This is certainly possible and has been done many times using the Sun as the lensing object, although the claimed accuracies of a short time-series of measurements on single sources is an order of magnitude lower and so probably wouldn't work for the Moon (e.g. Titov et al. 2018 ).
The Gaia astrometry satellite has likely end-of-mission positional precisions of about 5 micro-arcseconds for bright stars, but it is unclear what the precision is for a single scan (which would be needed because the Moon moves!). However, to counterbalance that, one could also average the lensing effect over many stars surrounding the Moon and of course average this for different stars with the Moon observed at different times over the entire Gaia mission.
Unfortutunately, although in principle this could be done, because of the way that Gaia scans the sky it is always pointing away from the Moon and so there will be no observations towards the Moon that would be capable of revealing these small deflections (e.g. see here).
Using this formula from Wikipedia, $$\theta = \frac{4GM}{rc^2}$$ and plugging in the mass and radius of the Moon gives a deflection angle of $1.2567 \times 10^{-10}$ radians = $2.592 \times 10^{-5}$ arc-seconds. In comparison, the deflection for the Sun is just under 1.75 arc-seconds, using the Sun's equatorial radius.
I expect that it would be very difficult to observe such a tiny deflection from the Earth's surface, even using adaptive optics.
Using our Moon as a gravitational lens might prove extremely challenging if not less than the angular resolution of positional measurements allow. It's a small effect with the Sun, and the Moon is 100 million times less massive and 100 times smaller in radius. The angular deviation from the Sun is less than one arcsecond and scales as M/R. So the deviation due to the Moon would be around $10^{-6}$ that of the Sun. Current technology allows angular resolution in astronomy of a few micro-arcseconds. In essence the deviation the Moon makes on light's path is about 10 - 100 times less than currently detectible with the best telescopes.
However, gravitational lensing can be used to actually detect exoplanets and exo-moons (i.e. moons around exoplanets).
The basics of that have been worked out around theoretically around 10 years ago (e.g. see this article). Meanwhile there is a few planets confirmed found by this method via the OGLE collaboration - I'm not aware of any exomoon found so far.
