Sister Stephanie Baliga isn’t a professional runner. She has no coach, no sponsors, no chance of earning a spot on the U.S. Olympic team. Her duties at a Catholic mission serving the poor leave her little time to run at all.

Yet on Oct. 11, Baliga hopes to finish the Bank of America Chicago Marathon fast enough to qualify for the U.S. Olympic Marathon Trials, to be held in Los Angeles in February. “If I achieve this, I will be completely satisfied in my running career,” says Baliga, 27, a former University of Illinois distance...