Specifically from the later Roman periods including the invasion of Britain.
How much of an impact would foraging be able to provide? If you have 1000 men to feed, would having units gather roots and berries and such make much of a difference? Obviously an army would need grains and animals with them before going far, but i have a hard time picturing someone saying "Great! A fruit tree! We're saved!"
There was a huge organisation dedicated to supplying the Roman army. Supply for expeditions would be organised from Rome in advance. Supply bases would be set up at strategic locations; food would be sent by ship to harbours near the area of operations, and Roman centurions would be assigned to ensure the unloading and further distribution of the supplies went smoothly. Additionally, client states and other allies could be ordered to provide supplies to Roman legions operating in their neighbourhood. If a legion was stationed in the same area for a long time (as happened often in the later Roman Empire), they would organise some amount of self supply from territories reserved for military use (including grazing of horses and raising of cattle). Additionally, local communities would be ordered to provide supplies.
How does foraging fit into this picture? During campaigns in enemy territory, foraging was useful as much to add to one's own supply as to hurt the enemy's supply. But this meant stripping the land of much or even all of its resources and wouldn't be sustainable for long durations (this is one of the reasons why Hannibal kept moving through Italy in the early years of the Second Punic War). Running a military campaign solely living off the land was dangerous as you ran the risk of not finding enough. If foragers had to be sent too far out looking for more, they ran the risk of being picked off by the enemy.
All in all, running a military campaign supplied solely by foraging was really only possible in fertile lands during the right time of the year, and even then it was not a good situation to be in. In all other cases - in less fertile lands, already ravaged lands, off season operations (even just camping through the winter) - some kind of supply would be needed and foraging would at best be a small addition to the soldiers diet, breaking the monotomy of the usual supplies.
Source:
Peter Kehne, War- and Peacetime Logistics: Supplying Imperial Armies in East and West, in: P. Erdkamp (ed.), A Companion to the Roman Army, Oxford 2007, p. 323-338