You'll have to consult the regulations that pertain to the state in which you hunt. Where I hunt, you can use a cap & ball revolver during the regular firearms deer season if it is at least .44 cal and barrel is at least 7". But you can't use such a revolver during muzzleloader season, because it doesn't have a solid threaded breech plug. You can buy single-shot muzzleloading pistols with a breech plug, however.
My point is that regulations such as this are very state-specific. Pick up a copy of hunting regulations where you'd buy your license, or go to your state's department of natural resources website to find a PDF copy of the regs. Once in the PDF, search on "muzzleload", "blackpowder", or some such word/phrase.