Prior to the Sengoku period, the recruitment of non-Samurai foot soldiers (who would become to be known collectively as ashigaru) was done in a fairly haphazard manner. Conscription was common, was that was not always terribly effective and the conscripted soldiers were often of a fairly low quality. Large amounts of ashigaru would be drawn from groups of men who would basically join samurai armies at time of war, drawn to the possibility of looting and pillaging. Many of these men were poorly armed and armoured, unless they were able to steal the equipment of a dead enemy. Desertion tended to be very high, as one would imagine.

However, the constant conflicts of the Sengoku period encouraged daimyo to recruit their armies in a more stable manner. A well-organized daimyo in the Sengoku period would have detailed knowledge of his land and the amounts of men his retainers - his samurai - could bring under his command. He would have arms and armour to provide to them (the armour was known as okashi gusoku, or "loan armour"), which would give an impression of uniformity throughout a clan's army. The majority of ashigaru were still part-timers - they were farmers who fought when they were called to fight - but the increasing effectiveness of ashigaru on the battlefield (things like massed archery and spear formations, which hadn't really categorized most past Japanese conflicts, become very important, and ranks of officers amongst the ashigaru emerge) shows that they were better trained than the ashigaru of before the Sengoku jidai. Some ashigaru would actually attach themselves to samurai and become full-time soldiers, something that was more common in better developed and more populous fiefs. Toyotomi Hideyoshi had large amounts of these soldiers, but Hideyoshi had, of course, far more resources than most other daimyo. Calling up the peasantry to arms was indeed the main method of filling an army's ranks.

To answer how this would occur, different retainers were obligated to provide so many men to their lords. I can't say that I know too much about the practicalities of these calls to arms, but a daimyo would likely send messengers out to his retainers, who would be responsible for organizing their contingencies of men (which could include samurai sworn to those retainers, ashigaru, and standard labourers). Prosperous landowners likely wouldn't conscript all of their peasants as ashigaru, because to do so would damage the economy. Takeda Shingen, a more prominent Sengoku daimyo, set up a system of fire beacons (think of the beacons on Lord of the Rings, if you are familiar with that series) and permanent messengers to allow for the news of an incursion into his territory to be spread quickly so that an army could be drawn up more rapidly.

How many ashigaru a daimyo could call upon, and how capable they were, was something that was very important in the Sengoku period. Popular history focuses on the samurai, but ashigaru were absolutely vital in campaigns. By the end of the Sengoku period, Toyotomi Hideyoshi and later Tokugawa Ieyasu had armies that were mostly professional in nature.

Source: Stephen Turnbull's Ashigaru 1467-1649. It is a fairly short book (a part of Osprey's Warrior series), but quite frankly there aren't many books out there about the Ashigaru and a lot of the information out there on them comes from Turnbull.