No, ammunition is very easy to make. If you've ever hunted or know someone who hunts, they don't throw away the wasted cartridges, right? They keep the casings and reload them for future handloads. If you know the right measurements, it's just a simple matter of filling, priming, and recapping. In a way, it's just a little bit more effort than, say, molding a lead ball, making gunpowder, and a percussion cap. All the above material used to operate a flintlock gun are now used in bullets, except in a more simplified form.
Plus, I think there wuld be an abundance of ammunition, even despite the war. It's not like they lack the facilities. The Gun Runners and the Brotherhood seem to be up and about, right?
Besides, in a world where just about everyone is totting a gun and carrying around stims, bullets aren't such a stretch.