To make stoneware pottery, you need stoneware clay. You could possibly use that clay to make a kiln to fire pots made of the same clay, but stoneware is vitrified clay, and that is a different ball game from earthenware.
On the positive side of things, I know from
experience that you can use local earthenware clay to make a kiln to fire pots made of the same clay, and it will last many, many firings.
Unless you are seriously into medieval reenactment I recommend a domed downdraft style. Superficially this is much like the kiln in the
video, but has a single central hole for the fire to enter the kiln and typically four equally spaced exit flues. Once the fire is going nicely the central hole in the top
should be completely blocked so the hot gases redirect to the exit flues. (think
rmh barrel, only without the insulated riser and with flues to encourage the draft rather than a bench.)