The clay in Graveyard Keeper is mandatory in some story missions; the second quest from Episcope sees you craft 20 clay bowls. Clay is also useful for repairing gravestones.
Unlimited deposits of clay
There are a few holes in the ground, from which you can extract clay - such as the one above. There are a few spots like that, but the deposits are unlimited. The nearest one is close to the graveyard.
What can be crafted from clay?
In order to create things from clay, you need the skill, Clay. Then, you can make the Potter's Wheel. You can place it next to your house and craft clay bowls. You will need clay and water.
You can use clay to create a Stone repair kit, used for repairing gravestones. Just use clay, simple iron parts and a piece of stone.