It appears that service records for the regiment were never recorded.

As a last resort, it’s possible, but I think unlikeable, that the regiments service records may have been summarized in this history,

A history of the 15th Missouri Cavalry Regiment, CSA: 1862-1865
Jerry Ponder - Author

https://www.amazon.com/history-15th-Missouri-Cavalry-Regiment/dp/0962392227