You need space elevator. And not only one. If my calculations are correct, you need four of them, possibly connected to each other.
(Googling Space elevator length gave me 60000 miles = 96560.6 km. Googling "distance to Moon" gave me 384400 km. $384400 \div 96560.6 = 3.98$, thus 4 space elevators)
You need to somehow resolve variable distance of Moon to Earth
As stated in the comments, the Moon changes its distance relative to the Earth by 10 percent. This means its distance can be:
$$(384400 - (0.1 \times 384400)) < \text{ distance } < (384400 + (0.1 \times 384400))$$ $$ 345960\text{km} < \text{ distance } < 422840\text{km}$$
So, you do not only need 4 space elevators, you need to build space elevators using curved cable, which is another level of the whole engineering issue.
You need to resolve losses on the cable
Being in energy field, I have to tell you sad news. The power cables lose their power on the go. We kinda resolve it by pushing the power on high voltage to the cables, but you are still going to lose some power.
Long story short, I believe it is more plausible story to beam energy than to send it via cable
