Can you sell found weapons in Breath of the Wild?
According to this source, it is not possible to sell weapons:
Nope. Weapons in the game looks likes a completely different kind of item. There's abundance, but you can't trade, repair (usually) or buy them. They're spoils.
The exception appears to be armor only.
Additional GameFAQs source and Reddit.
The NPC who gives you the side quest to get a piece of Royal Guard's equipment will also trade one gem for each piece of Royal Guard's gear you bring her. There's no way to sell regular weapons, though.