The firm now hopes to work with human volunteers, perhaps as early as next year should America’s Food and Drug Administration play along.    

 should America’s Food and Drug Administration play along

= if America’s Food and Drug Administration should play along. 修饰不定式to work

if 引导的条件句如果含有were, had 或 should, 可以省略连词if，将were, had 或 should 倒装于主语之前。