My Refrigerator Water Tastes Like Peppers!
My refrigerator also has as a through the door water dispenser like yours. Inside the refrigerator behind the bottom produce drawer there is a large coil of plastic tubing that is designed to pre-cool the water that is delivered to the door spigot. 

You should check your refrigerator to see if it also has this same tubing arrangement. It is entirely possible that the previous owner of the refrigerator had left vegetables in the drawer for such a long time that they became severely spoiled and contaminated this tubing. Contamination could also have happened if spoiled food was on a shelf and decay liquids ran down the back wall into the tubing area. If the lower drawer was cracked and leaked out into the bottom of the refrigerator this problem could have been even worse. 

If your refrigerator has this system I would highly recommend that you simply replace this coil of tubing with new.