There are a few possible answers here.
First, the world wasn't completely devoid of competent individuals. Ragnar Danneskjöld had found competent people to trade with even in the various People's States. There were a large number of at least somewhat competent people in the country, too; for example, Minnesota had produced
The gangs of unemployed that were roving the country actually included a fair number of competent people. For example, Jeff Allen (a bum who hitched a ride with Dagny who was able to tell her who John Galt was) was evidently competent (although he had been suffering the consequences of his involvement in the fiasco at the 20th Century Motor Company). Also, Hank Rearden dealt with a gang of unemployed that was described as follows:
Rearden, that evening, his coat collar raised, his hat slanted low over his eyes, the snow drifts rising to his knees, was tramping through an abandoned open-pit coal mine, in a forsaken corner of Pennsylvania, supervising the loading of pirated coal upon the trucks which he had provided. Nobody owned the mine, nobody could afford the cost of working it. But a young man with a brusque voice and angry, dark eyes, who came from a starving settlement, had organized a gang of the unemployed and made a deal with Rearden to deliver the coal... Watching him at the job of loading the trucks, Rearden thought that the boy, if born a generation earlier, would have become a great industrialist; now, he would probably end his brief life as a plain criminal in a few more years.
Even the bum that Eddie Willers encountered in the beginning of the book appeared to be intelligent.
The primary problem in Atlas Shrugged wasn't that none of the Looters had ability (which many of them admittedly didn't), it's that they refused to exercise it. In particular, the "cardinal sin" of objectivism is the refusal to exercise your mind. Clearly, not everyone has equal ability, but everyone has the same moral capacity. For example, Francisco D'Anconia told Hank Rearden that, while Eddie Willers couldn't have invented Rearden Metal, he could match Hank and Francisco in morality. Presumably, if enough of the Looters were willing to change, they could become competent (either relatively quickly due to the fact that they were actively using their existing ability and intelligence, or over time because they were learning and growing over time).
Second, both the Strikers and the people who quit after hearing John Galt's speech were already producing enough for themselves.
Third, even relatively late in the Strike, there were still people who were able to produce. Many of them were unable to actually produce or deliver their goods for a variety of reasons (e.g. the harvester company that "went under" because they couldn't get steel, the Minnesota farmers who couldn't deliver their harvest because Taggart Transcontinental's trains didn't show up, etc.).
Fourth, many of the people who were working but had been forced to "hide" their abilities (e.g. the Taggart Tower Director) would still be available.
Fifth, there are still plenty of people whose main problem is cowardice, not incompetence. For example, the text doesn't actually say that Mr. Mowen was especially incompetent as far as I know, just that he was a coward who was afraid to continue to use Rearden Metal after the State Science Institute's statement on the matter. He presumably could have been at least reasonably good manufacturer if he changed.
Finally, there were many competent people that had simply abandoned their jobs (such as Pat Logan and the rest of the crew that was on a particular run of the Comet). It's unclear exactly what had happened to them, but they may have become available as well. (This is especially the case given that the main reason they abandoned their jobs in the first place was Directive 10-289).
