Some Pokedex entries state Farfetch'd can eat their stalks in dire situations, search for new ones, and some fight over sticks. This makes me think it's bs Pokemon logic of Pokemon with objects/multiple units being born with those objects/units (think exeggcute, kangaskhan, timburr, sinistea, possibly charcadet), but they can find new ones if they need to. Plus, there are leek items in game, so it's not absurd to think that Farfetch'd can find a new leek if they need to.
As for leek season, there's no mention of such a thing in game, but I'd like to imagine that even if there was, Farfetch'd will somehow always be able to find one.
