There is plenty of material....
The orbital radius of the moon is $3.85\times10^{8}$ meters. In order to block all sunlight from the Earth, we would need to transform the moon into a solid shell (basically, like one of the Halo rings). Since the Earth's axial tilt is significant, the Moon-shell will need to extend roughly 23 degrees north and south of the Equator.
The surface area of a sphere is $4\pi r^2$. The surface area is a polar cap is $2\pi r^2(1- \cos \theta)$, where $\theta$ is the polar angle of the cap. In this case, that polar angle is $\theta = 63^\circ = 1.1 \text{ radians}$. Therefore, the overall surface area of the shell-like debris field must be
$$4\pi r^2 - 2\cdot2\pi r^2\left(1 - \cos\theta\right)=4\pi r^2 \cos\theta.$$
Plug in the radius and polar angle and we get $8.4\times10^{17}$ m$^3$.The mass of the moon is $7.3\times10^{22}$ kg. If you spread the entire moon out over the required area, then you have about 86401 kg available to block the sun for every m$^2$ of surface area.
The moon's density is 3300 kg/m$^3$; but that material is somewhat compressed by gravity. Lets say the moon's density is 2500 kg/m$^3$ when not in a gravitationally compressed ball. This is slightly less than the density of the Earth's crust, so probably not a bad estimate. If we use that figure, then you can form a solid shell 34 meters thick around the Earth. That will definitely block out the sun.
If you decide to only block 1% of the sun's mass into space; there is still enough for 30 cm of solid rock and iron (over a foot!) blocking the sun from the entire surface of the Earth.
...but it will not be in the correct place...
This is going to be a big spherical cow estimate, but let us say that the Moon is destroyed completely and turns into a structure the shape of Saturn's C ring (the innermost bright ring). A mass of $1\times10^{18}$ kg is the distributed around an inner circumference of about 135,000 km. This is four order of magnitude less massive than the moon; but then twice as close to the main planet. Also relevant is the density; Saturn's ring is water ice but the Moon is about 2.5 times more dense, as estimated in the previous section. Taken all together, let us estimate that the Moon turns into a ring with length dimension $10000\cdot\frac{1}{2}\cdot\frac{1}{2.5} = 2000$ times larger than Saturn's C ring.
Unfortunately, Saturn's C ring is only 5 meters thick, so our Moon ring is estimated at only 10 km thick. To cover the 23$^\circ$ north and south fo the equator, it really aught to be 300,000 km thick. That isn't going to cut it. A mere 10 km is going to block less than 0.001 % of the light blocking Earth; this won't have much effect at all on the climate.
Conclusion
There is plenty of mass, but naturally forming ring structures are nowhere near thick enough to block the Sun; certainly not at the distance from the Earth that the Moon already is.
Unless you engineer a solution, like blowing up the moon, moving all the rocks into a closer orbit, and then using some mechanism to disperse them from a naturally forming ring, you simply aren't going to get appreciable blockage of the sun.
