How to add electrical fixtures to (custom) furniture?
Assuming your furniture has Locations X and Y where you'd like to mount the receptacles, and location Z that would be a suitable, safe place for a power cord to exit the furniture without being underfoot or hazardous: 

To connect X, Y and Z, I'd use steel or plastic junction boxes with steel or plastic conduit between them. Preferably steel (though it's harder to work with). Route between boxes doesn't matter.  You can go with plain old ugly stamped boxes or they make some fancy cast ones with threaded ports.  Whatever works. 

At location Z, I'd have a flexible cord with proper grommets and strain reliefs.  This is "cordage", or cable specifically made to be flexible equipment cords.  (obviously cordage is common since every appliance has it; but you must use cordage, not random Romex or whatever.)  I would use a minimum of 14 AWG, preferably 12 since it will assure a breaker will trip before it overloads. 

Inside the conduit, I'd see what makes sense.  If I can spare the length, I might pull the cordage through whole. Otherwise I'd splice it to THHN single-conductor wire run in the conduit between the boxes.  Either go Z then Y then X, or branch Z-Y and then Z-X.  Either way is fine. 

I would NOT staple Romex between the junction boxes, because Romex is vulnerable to physical damage, that being the point of the conduit.