Could anybody tell me from where can I order a spare door for the battery of my camera, a Sony NEX F3. I need to replace it as it was broken, and I cannot find it online.
As a mere mortal, you probably can't. Selling parts is a hassle, and additionally most companies protect their authorized service centers in this way. So, your choices are probably one of:
Send the camera in for repair. If the door came of as part of normal use, I would expect that to be covered under warranty.
Look out for broken cameras of the same model on eBay or other online auction sites. People sometimes try to get a few dollars out of electronics that don't work, and maybe you will get lucky there.