From the above Wikepedia link:
The Colt Walker was quite powerful, with modern replicas firing modern FFFg black powder producing energy levels in excess of 500 foot pounds with both picket bullets and 0.454-inch-diameter, 141-grain round ball bullets. The black powder Colt Walker has a muzzle energy nearly exactly the same as a 4-inch-barreled handgun firing a .357 Magnum
I've seen a deer taken down with a 4" .357 and it was impressive. And Illinois requires a minimum of 500 ft/lbs at the muzzle for handguns. So a Walker would do it (I'd keep it to bow-hunting range shots), but prolly not a 3rd model Dragoon.
(I only linked Illinois regs 'cause they're the first ones I found during a quick search)