For Akuma you can use continues as much as you want. However, when you get a game over, the counter for perfects in order to fight Sagat will be set back to zero. I guess this is the case to prevent farming perfects in the boss rush by beating the easier ones again and again.
Therefore, you can only use continues before you get your very first perfect towards Sagat (and, of course, when you finally have reached the Sagat fight).
Also notable is that Balrog and Vega count towards both Akuma and Sagat.