The stuff you'd need to stockpile is unlikely to deprecate in value and it certainly isn't perishables, so doing so really isn't a big deal.
Also once you get an order you already know roughly how much steel you'll need - a bit of lag as you wait for the first batches to arrive really isn't a big deal in such a project.
Presumably you'd get everything lined up before you finish your previous project. Hardly any construction just simply starts the day someone approaches a manufacturer anyways.
Relevant to US shipyards, domestic oil can only supply domestic refineries if there's transport, and there's not enough oil carriers that satisfy the Jones act, because they need to be US built, US flagged, US owned, and US crewed. If the best way to transport the crude is by ocean, it's not going to be extracted and refined in the US; it only makes sense to do both in the US if crude is delivered by pipeline or rail. There's also issues with the grades of oil and how refineries are setup, but that's more tractable if there's supply.
A slightly different version of this which does carry some weight - an oil producing country could use an export ban to lower domestic prices.
Just because steel is made locally doesn't make it any cheaper. Transport costs are very small these days, thanks to modern shipping. So a local steel producing isn't going to sell below the worldwide market rate, which is defined by the largest steel producers and consumers on a global scale. I remember similar debates years ago about oil, that if the US produced enough domestically that imports could stop and local prices drop. That isn't how the modern world of free trade works.