That looks to be a .32 Double Action 4th Model, from around 1900. Chambers 32 Smith & Wesson cartridges. That is the short cartridge not the 32 Smith & Wesson long.
Guy
Try a big box Sporting Goods store like Academy Sports, Cabela's , Gander Mountain. If you don't have one of those near you, you may have to order it off the internet.
Guy
Any brand commercial ammo that you can find should be fine in the gun.
Guy
Welcome! Here is one source:
Magtech, .32 Smith & Wesson, LRN, 85 Grain, 50 Rounds - 85430, .32 S&W Ammo at Sportsman's Guide
If your .32 DA is in good condition it will handle this fine. In the right light, the bullets move so slow you can actually see them in flight (no kidding!)