Technically nobody can tell you if you can do it since nobody knows your electrical system. There are rules in the electrical code that dictate how many devices you can have on one circuit, so although you can hook up a wire and add a receptacle, that doesn't necessarily mean it will meet code.
Also you need to be concerned with whether or not the bathroom receptacle is GFCI protected. It won't hurt to plug your vacuum into a GFCI receptacle of course, but if that GFCI is feeding other outlets already you may not be able to tap into the feed side of it to add another outlet without overloading the box with too many wires (again there are rules in the electrical code for wire count etc).
In theory assuming everything meets code this is a pretty easy change. Just be sure you use an old-work / reno box and that you verify there is no obstructions in that particular stud bay which could cause problems (plumbing, HVAC ducting etc).
Before cutting your hole for the new box, I suggest removing the cover plate and receptacle you want to add on to. That will give you more room to fish the wire down as well as allowing you to see what side of the stud the box is mounted to. The last thing you want is to cut a hole and find out you are on the wrong side of the stud because fishing a wire through a stud and then up to your old receptacle is much more difficult and will take a lot more time.