They had it, but it wasn't mass-produced, and nearly all doses went to soldiers on the front that had suffered injuries or amputations. There wasn't enough for ordinary infections - they'd even collect the urine of soldiers being dosed so that excreted penicillin could be distilled and re-used.

I figure that the story took place either during the war or slightly after (late 1940s). Penicillin wasn't mass-produced and available to consumers until the 1950s.