There were some Lemats supplied with flasks.

Krider of New York made at least two Lemats as models. The reciept for his work on one of these includes a charge for a flask...apparently made from scratch rather than a commercially available flask.

Also, Albaugh's "Civil War Handguns" shows a picture of a cased pair of Lemats with flasks. It appears to me that the body of these flasks are made of ivory.

The vast majority of Lemats, those made under contract with the Confederate Army and Navy, would not have included flasks. At least there is no evidence in the records, as far as I have seen, of the contracts including any accessories other then ammo and percussion caps (and those only with the Baby Lemats for the Navy).

Typically, particularly by the time of the Civil War, flasks and molds were only used/supplied with cased revolvers. In practicality, few users were pouring their own powder or making their own balls. By this time, cartridges (powder and ball packaged together as a unit in a paper wrapper) were widely available.