It is possible, wardrobe people probably wont do it, you will probably need an electrician, it depends o the type of wall, if it is a stud partition wall may be easy to run a cable behind, and connect up new socket where you want it, then plaster over old hole.
cost depends on electrician, they will charge a callout fee on top of job, would probably be worth your while to see if there are any more jobs you want them to do as well.