I know that 'living off the land' or 'foraging' are just generally euphemisms for stealing food from local farmers, but I'm wondering how organized the process was.
Did they just let all the men loose and tell them to figure it out themselves? Did they organize it so every squad went to a different farmhouse? Or were there there dedicated troops for this, and the majority of men on the march were expected to stay with the main body?
And how large an army could realistically support themselves this way?
There are a few variations, but we have a good resource for how supply was managed in Carl von Clausewitz's On War, specifically one of his more neglected chapters in the middle on supply.
Living off the land is an old practice, but it developed over time. Throughout the early modern and Napoleonic periods, it often means literal foraging; armies, upon stopping for the day, would disperse with sickles to gather up grass for the horses.
In the wars of Louis XIV and Frederick the Great, armies would levy contributions on the territory, and if the community failed to furnish the supplies, they would carry out executions, essentially torching the villages. During the Thirty Years War, armies reserved the right to exact 'burning money'; if the army claimed it had a right to burn a given building under the laws and customs of war, the owner could give them a monetary payment to avoid this. This money would then be used to purchase supplies.
There are four basic methods of supply in the Napoleonic period.
Clausewitz recommends the third method most highly; local authorities notified in advance can gather supplies for a column from across a wide area using transport by the local population. Just as a ballpark number, local authorities can draw for supplies on an area of 100 square miles given a single day of notice; 400 square miles with two days of notice, and 900 with three. The advancing column needs only to collect the supplies at the designated spot, often paying cash for the supplies when possible. Armies will form detachment for use by the local officials. Living off the land was a well organized phenomena during the Napoleonic Wars; the words 'quartermaster' and 'logistics' actually come from officers who would determine where the troops would be quartered or lodged (difference is just whether you use the French or German term). Columns would send detachments of officers ahead of their advancing units to meet with the local authorities, carrying lists of their men and their supply requirements, and the local authorities would tell them how many buildings suitable for lodging they had, and how much they could supply.
This method is not iron clad, and isn't as thorough as foraging by the troops themselves, but accomplishes the supply of a large army with a minimum of fuss if it carries a few days of supply with it as a cushion. Essentially an expansion of the contribution system, communities can be threatened by being held responsible for shortfalls, either by direct punishment or by resort to foraging by the troops themselves.
He also mentions that the first method has the advantages of being extremely quick and not needing transport, but it presupposes that the troops will be quartered in households, who will furnish supplies for the soldiers there.
The second method, requisition by the troops themselves, usually gets a higher yield, as soldiers intent on finding food will clean out the afflicted community. Indeed, in the Napoleonic period, French units will pull out the best foragers in their units, the men who can find those hidden goodies, and make them the unit's designated foragers.
In terms of how many men could supply themselves this way, quite a lot. The numbers Clausewitz uses (again, ballpark, but well informed by study and experience) indicate that a force of 30,000 men can be supplied by a moderately peopled area of ten miles by ten miles quite easily, presuming it hasn't been previously occupied. Each house will have about a week's worth of bread, and you can expect about 70-80 houses per square mile in a moderately peopled region. Three such columns side by side can fight together on the battlefield, for a subtotal of 90,000 (which will probably equate to 75,000 fighting troops). This force can be followed on the next day by another force of 90,000, and suffer no privation; a total of 150,000 men is a substantial force. Even if 10x10 mile columns aren't enough, 15x15 mile columns more than double the area an army can draw on, which will usually meet their needs.