Is it possible to make go bowls out of clay? More importantly, is it practical?

You have to test it out. Normal fired clay would scratch stones, but if you are careful and apply lacquer I am sure it would be fine.

a) The surface hardness of a pottery bowl would be the hardness of the glaze, if any, rather than the harness of the fired clay. And the harness of that varies greatly by type.

b) Harder materials scratch softer materials. Slate and shell might be scratched but glass stones and probably yunzi stones harder than any pottery.

Certainly possible.
"Practical?" is answered by your application. If you were going to line the bowls with, say, leather or thick silk, or you were to leave a cloth bag in place, you could keep the most precious slate and shell stones in these fired clay bowls. Cheap glass or ceramic stones would get dinged up but that's why they're cheap. If you have sets of the weighted Ing stones, clay bowls could be an elegant step up.

But why clay? Will you decorate the sides and lids? Are you using special types of clays or bone china compounds? Are your throwing these, forming them from slabs, slumping, coiling or carving?
If you have skills as a ceramic artist, those skills might be adapted to other media. Among my go bowls is a pair woven of pine needles. they are delightful.

_________________
David Bogie, Boise ID
I play go, I ride a recumbent, of course I use Macintosh.

I just wanted to have something a little different. For example, I'm making a set of stones out of granite.