Reading up on Stone Mortar rounds, and now have a better idea of their ammo. So, according to the 'To The Sounds of the Guns' blog which is my general source for artillery knowledge, the rounds in stone mortars varied considerably. Since the mortars were intended either for siege or garrison troops and were quite limited in range and employment, they were really rare. Most were simply 18th century weapons soldiering on in their niche use because there wasn't a pressing need to replace them. Stones in baskets was one option for rounds, but not necessarily the preferred one. Rather, more prepared rounds using grenades (the sort used by 18th century grenadiers, like the ordnance corps 'flaming bomb") or canister\grape might be used. The old 18th century manuals said something to the effect of the stones used being "as round as possible" which to me sounds less like they were actually shaping rocks and more like they were just picking up any old round rock they could find lying around and throwing it in a basket! So really any sufficiently round rock is a stone mortar round.
Because the Stone Mortars were so rare (I think a grand total of 2 were ever actually designed, ordered, and cast by the US, the rest we just mortars we sort of inherited), Coehorns got used to fill the role. There would have fired grape shot. Yes, actual grapeshot, not canister. This likely was because grape was available in the 24 pdr caliber, being that the other guns in the system were seacoast guns and would have use for such a thing. And the forts most of the Coehorn *** stone mortars lived at would have had this naval round in abundance.
So clearly, sir, what you need is a stone mortar, and a basket to fit it. Then all the worlds round stones are Civil War artillery rounds for your finding!