What is the meaning of 庸 in the phrase 毋庸置疑?
庸 means 需要 (need) here.


  (1) (会意。小篆字形。从用,从庚。“庚更”同音,表更换。先做某事,然后更换做别的事。本义:用;需要)
  
  (2) 同本义 [need;adopt]
  
  (4) 常与否定副词“无”、“勿”、“弗”连用
  
  (5) 又如:无庸讳言;无庸细述;毋庸讳言;


So 毋庸 means 无须/不必 (no need to).