Half, Late, and Full Moon Grass can be purchased from Patches, the Hyena in the Nexus after unlocking him in 2-2 or 4-2. At the end of the game (when the organ is playing), you can purchase New Moon Grass from him. His prices on these grasses are all reasonable, so you may find it more productive to simply farm for Souls instead, and buy the grass you want from Patches. He also sells some Dark Moon Grass near the end of the game, but the quantity is limited.
Crescent Moon Grass
If you have unlocked the execution grounds in 1-1 (requires either PBWT or PWWT), each of the BP Dreglings drops Crescent Moon Grass.
Note: These are pretty useless at higher Soul Levels, but can be used to maintain the effect of the Clever Rat's Ring and the Morion Blade.
Half Moon Grass
The Blue Eye Knights in 1-1 and 1-2 all drop Half Moon Grass.
Warp to the Tower Knight Archstone and head back to the bridge. There are two knights standing there ready to be backstabbed.
They also drop Late Moon Grass sometimes.
Full Moon Grass
In 1-4, the Red Eye Knights drop 1 - 3 Full Moon Grass every time you kill them. They net you a decent amount of souls per trip as well. They sometimes also drop New Moon Grass.
You can get as much as 6 Full Moon Grass per kill.
From the Penetrator Archstone, go backwards into 1-3 and kill the three Red Eye Knights there, as they will drop Full Moon Grass.
Dark Moon Grass
The Depraved Ones in the 5-3 boss room will drop them. Additionally you might get them from the Depraved Ones in the shanty town just before the Dirty Colossus Archstone; at the very least they'll drop New Moon Grass.
You may also be able to get them from the Depraved Ones throughout world 5, but it's easier to get them later into 5-2 and in 5-3.