Is it possible to recover precious materials from a weapon to use for crafting other weapons?

The Crafting rules are vague

The section on Crafting with Precious Materials section, as you may have read, only says

When you Craft an item that incorporates a precious material, your initial raw materials for the item must include that material; at least 10% of the investment must be of the material for low-grade, at least 25% for standard-grade, and all of it for high-grade. For instance, a low-grade silver object of 1 Bulk costs 20 gp. Of the 10 gp of raw materials you provide when you start to Craft the item, at least 1 gp must be silver. The raw materials you spend to complete the item don’t have to consist of the precious material, though the GM might rule otherwise in certain cases.

It doesn't specify what state the material needs to be in so it is reasonable, and within your power as the GM, to decide for your game.

As Game Master, it falls on you to adjudicate the rules. This means you’re making judgments and decisions about the rules, especially when their application is unclear.

You have the ability to say using a low-grade silver longsword is viable "raw materials" for creating new silver weapons. Furthermore, there is real-world evidence (reddit) that weapons could be re-smelted, so it makes sense for this to be the case.

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As an aside, it's worth noting that the only benefit to allowing this is "access to" the silver material. Silver, unlike some other crafting material is not Uncommon and "should" be available most places (this is, of course, up to the GM). Mechanically, the party is better off selling the equipment to the town guard or a smith looking for material and using that money to buy exactly how much silver they need due to Crafting's efficacy being nearly identical to other Downtime activities' income rates.