Is it possible to mount a derailleur on this bike?
Sure.

Buy an adapter claw like this one (e.g. Sunrace SP550.):



The claw screws into the dropout, then you bolt on the derailleur as normal. 

Alternatively, you can buy a derailleur with an integrated claw like several in the Shimano Tourney line (E.g. the Shimano Tourney RD3105; actually pictured, RD8153):



Note that you still have the issue of getting in a wheel with multiple cogs in the frame. This might require spreading the frame, if its steel (not a good idea on non-steel frames).  

As mikes and mattnz point out in the comments, you also have to add cable stops and/or cable guides to the frame to route the shift cable. Or, run full length housing and use zip ties.  Problem Solvers among others make clamp on ones, and Origin 8 sells some zip tie-based cable guides you can use instead. You'll need to do a bit of searching and work probably to get the right routing, but they make cable stops that clamp onto the chainstays, the top tube, bottom tube, etc. A shop should be able to sort you out with these things relatively quickly. Other options might be to braze/weld on some.