You might get a better response to this question if you picked a specific army and campaign. The Hundred Years War and the Crusades are quite different.

I will throw out a few generalities however.

Firstly, Medieval armies hardly ever needed to be resupplied with weapons. They fought relatively few pitched battles where they would need a supply of arrows or the like. Generally armies carried enough missiles to fight one big battle, and that would decide the campaign. English Hundred years war armies that relied on the Longbow had a slightly more complicated arrangement, but compared to the ammunition a 20th century army would use it's still pretty rudimentary.

The most difficult article for a Medieval army to obtain, by far, was fodder for its horses. The reason is that fodder is very bulky, and therefore difficult to move by cart (before the carthorses themselves consume it.) Even infantry armies need quite a lot of horses for various tasks and horses eat quite a lot by volume. Aside from fodder, food is the main quantity medieval armies need to supply. Lastly, in some situations obtaining timber for siege engines was a logistical issue.

To a great extent, moving armies before the 20th century "lived off the land" to a degree that seems astounding to people who study 20th century logistics. Because fodder is so difficult to move, a mobile army almost HAS to obtain supplies locally. Carts simply can't keep up. "Living off the Land" is in quotes because it is a complex and many faceted concept. There is not one way that this might occur. It might be military leaders sending polite letters requesting supplies to towns on the route of march and offering payment... it might be outright pillage, and it might be almost anything in between. If for any reason an army had to cross a barren, lightly populated area hardships would follow quite quickly. The First Crusade's movement thru Anatolia is a good example.

"Living off the Land" becomes immediately problematic once any army stops moving. The local population generally becomes unable to supply an army very quickly. In friendly territory, getting enough food and fodder to an army becomes very difficult. In enemy territory, it becomes extremely difficult. This is where sea or river connections become paramount. Supplying a siege by land is a very difficult feat until railroad connections in the 19th century. This is why in the first crusade Antioch could be besieged for months(although even with sea supply, most of the First Crusade's horses perished at Antioch) , from October to June, while Jerusalem had to be quickly assaulted in a month or so. Staying in the same place in hostile territory longer than a month would induce starvation, first of horses, then of men.