Caps,
actually neither one would have been a common item in 1866. Shotshells would have been carried in pockets, an over-the-shoulder shell bag or haversack type of carrier (Upland Bag), or a belt pouch.
That being said, it is FAR easier to pull shells out of canvas loops than leather.
If you're thinking about competition at a match other than an NCOWS match, I'd go for the canvas belt.
Many at NCOWS matches use either military cartridge boxes on the belt, or the shotshells slipped into the lower vest pocket, positioned upright. They stay put there and make for easy retrieval two at a time.
If your era is a bit later, into the mid 1870's onward, then modified 'prairie belts' like Cabela's would fit in. Into the 1880's and later, the Mills-type as used by the military would be popularized. By 1900 you would see both types offered in the catalogs like Sears, Montgomery Wards, etc.