In the immediate postwar years, the Japanese economy was in shambles. All major cities except Kyoto had been extensively bombed (the firebombing of Tokyo in March 1945, though perhaps less existentially horrifying than the atomic bombing of Hiroshima and Nagasaki, was probably responsible for more death and destruction than either of the latter), leaving factories, infrastructure, and supply chains in total ruin. Production facilities elsewhere were largely geared for war materiel, and though they could theoretically be retooled for peacetime production, such expertise was rare and the needed capital was scarce. Lean years of starvation had left the countryside largely picked clean. And of course a huge percentage of working-age Japanese men were dead, injured, or still awaiting repatriation in the former colonies overseas – a process that would stretch into the mid-1950s. Add to all of that the American occupation army from September 1945 throwing everything into disarray. In fact, it was the unofficial policy of the Americans in those first few years to concentrate on building democratic institutions and civil society while deliberately neglecting the Japanese economy. This was a punitive measure: public opinion in the US was that the Japanese "deserved" to suffer the hardship they had brought on themselves.

So, amidst this chaos and wreckage, how was your average Japanese city-dweller to survive? The answer of course is the black markets (闇市, yamiichi). These sprung up like weeds in every city and town – endless rows of cobbled-together stalls, buying and selling everything imaginable. They operated in a legally gray area, as far as licensing and zoning and all that goes, but the SCAP tolerated them, understanding that they were vital to Japanese survival, the key way that e.g. produce from the countryside made it to city tables.

And naturally, being of ambiguous legal status, the black markets were a hotbed of organized criminality. The usual gangster stuff was all going on: protection, kickbacks, counterfeiting, scams, contraband, numbers running, etc. There was fierce, often bloody competition for turf in this period. Old yakuza families and new criminal syndicates (often composed, as you suggest, of former soldiers, who sometimes had access to stolen or even simply forgotten military supplies) fought over who controlled these insanely lucrative markets. The winners, flush with capital, then turned to the even more lucrative business of postwar reconstruction: debris removal, building roads, sewers, apartments, factories...pretty much everything. By the 1950s the yazuka were firmly entrenched as some of the key players in the modern Japanese economy, and that has largely remained true to this day.

For a good summary of the immediate postwar, check out John Dower's Embracing Defeat. If you're interested in dramatizations of this period, I'd check out Pigs and Battleships (1961), Gate of Flesh (1964), and Battles without Honor and Humanity (1973). Although not about yakuza per se, extensive sections of Stray Dog (1949) were surreptitiously filmed in the actual Ueno black market, giving it a real documentary feel.