Do I need a permit to ground an outlet?
Is the box grounded?

Just because it's metal doesn't mean there's a valid grounding path back to the service panel. They often used metal boxes simply because they're more durable and better able to contain thermal energy from a short. 

If the box isn't grounded, don't install a 3-prong plug, unless it's a GFCI with "No equipment ground" label. 

You don't need a permit to swap a receptacle

And attaching the ground wire is part of replacing a receptacle.  No big.  

If that box is grounded, it may have a single hole tapped for #10-32. They make special ground screws for that (with or without pigtails), or any 10-32 screw will do. You can tap your own hole but it must be #6 or larger and -32 or finer thread pitch, in order to engage enough threads.  Using a sheet metal screw is right out.

There's a chance the receptacle is already grounded through the yoke. 

The yoke is the metal outer frame, that the mounting screws go through.  If the box is grounded, AND the yoke is run down hard against the steel box (with no plastic bits in there), that's a legal ground path.  

It's not a legal ground if the ears are holding the receptacle above the box e.g. against drywall, and there's some length of screw thread still unbottomed. 

It's also not legal if paper, plastic or paint is trapped between the box and yoke.  This often happens from the square of paper/plastic they put the screws through to "capture" them.