I think it depended on context for which.
I would imagine most of the time, an archer on land would carry their own bow with them unless it was an impediment (like, say, if they're off duty but part of a town or fortress's garrison and only take their bow when on duty).
Medieval warfare was for the most part Not pitched battles; these were relatively rare. Rather, raids and counter raids (called chevauchée) into lightly defended territory, ambushes, and sieges, were much more common. And even when there were pitched battles, there was often a town or city or fortress as the main objective, and the people trying to defend the place would offer battle as an alternative, usually in an attempt to relieve or prevent a siege. Pitched battles did start to become more common in Early Modern times though with increasing numbers of paid, professional, mercenary or civic standing armies, rather than temporary levies or mercenaries or militia that were relied on most of the time in medieval armies.
In addition to garrison duty in a town or fortress (where only those on watch would be armed all the time), or serving aboard a ship (where you could store arms away until needed as surprise attacks on sea were rare, you'd usually see them coming), or being part of an official military raid/mission, it's likely that there was certainly some way to resupply if they ran out of arrows, but that doesn't mean they never carried their arms with them.
Archers might also be in small groups foraging, or laying in ambush, or any number of tactical situations that would make resupply of arrows difficult, indicating that unless the archers were part of a large force on the march, their arrows would probably be with them, and their bows too I would imagine, unless the situation didn't call for them to be on alert (like if they're far from the front lines or in a garrison and off duty).
There were also independent forces using archers, usually bandits that preyed on local populaces, and the more formidable groups had seized a key fortress. They would more likely carry their own equipment than groups connected with a particular lord or faction.
A related example where not carrying your equipment with you turned out poorly was at Crecy, where the Genoese crossbowmen didn't have their pavise shields with them (left in the baggage train, and their crossbow strings were wet) when trying, and failing, to engage the English archers.
