Most games that I have been in haven't bothered to track mundane ammunition quantities.
If you are going that route and tracking things that finely, I would probably allow picking up and slinging appropriately sized and weighted items as an improvised weapon. Assuming that the character can find them.
I wouldn't consider a silver piece to be an appropriately sized and weighted item to be used as a sling bullet. A silvered sling bullet is a regular sling bullet with a silver plating. Just like a silver sword is normally a steel core with silver plating. You could make a fully silver sword using high grade silver ingots, but it has quite a bit higher of a price if I recall correctly. I guess you could do the same for sling bullets, but it would again have quite a bit higher price than is normal.