There's technically zero waste/scrap unless someone is failing their Craft check by 5 or more, but in that case half the raw materials is wasted/unusable.

When making steel there's a lot more than just the raw materials. You need fuel for the forge, for one thing. Steel is made by combining iron ore with carbon (coal dust, burned animal bones, etc.) and including a bit of glass or similar that the impurities will stick to. The whole thing is melted in a very, very hot furnace, sometimes taking an entire day to get it hot enough to melt the metal so it combines properly. Once that's done you have an ingot that can be hammered into a flatter sheet, which is then shaped and cut into one or more pieces of the armor. Any pieces cut away are added to the crucible for the next ingot. You wouldn't be finding any scraps because they always get used.

In modern machining there's a lot of cutting and drilling, resulting in a lot of chips and scraps that will be recycled if large enough to collect, or swept up and thrown out. In older times a saw wasn't used to cut metal, a tin snips was used which doesn't really result in any type of scraps or chips.

When a blacksmith strikes the metal with a hammer, the sparks you see comes from the impurities being knocked out of the metal. No useful chips or shards result from that.

There's basically no way to accomplish what you've suggested. You're not going to backstory a free suit of armor.