After some more internet research, I think you could make hand-reloaded bullets for modern firearms, but you'd have to have access to sulfur, which could be difficult for some Apocalypse Worlds. On the other hand, if you have volcanic activity or hot springs, it could be easier. It works in my AW because there are still-operating oil rigs with mini-refineries that can extract sulfur from crude oil (and also produce gasoline and kerosene for my setting).
So it all starts with urine (naturally). Ferment the urine for a couple months in buckets with a lot of straw in them. Chemical salts accrete on the straw. After months of this stinky process, take out the straw and wash it in water. Pour the water through wood ashes, then let it evaporate and it will leave you with saltpeter.
Burn the saltpeter with sulfur in the presence of steam and then mix the result with water and you get sulfuric acid. Put aside 2/3s of that.
Mix the remaining 1/3 of your sulfuric acid with more saltpeter and heat it and you get nitric acid.
Bathe cotton in 1/3 nitric acid and 2/3 sulfuric acid and then wash it multiple times in cold water and you get nitrocellulose (gun cotton), the primary ingredient of modern smokeless powder. The other significant ingredient is nitroglycerin, but that's also made with sulfuric acid and nitric acid.
After that, it's just a matter of casting new bullets in molds and using some fairly common tools to fill used bullet casings with your manufactured powder and fix them to the bullets. I'm sure it wouldn't be a totally safe process, but it's interesting that it could be done with pretty ancient processes.