You won't see any fault codes if the alternator is faulty. If you are measuring the battery voltage when the car is running and only seeing something like 12.5 to 13.3v then the alternator may be putting out some charge but not enough to keep the battery topped up. To charge a battery and maintain it's charge you need at least 13.7v and more like 14V across the battery.
A duff diode in the rectifier pack on the alternator will drop it's output quite considerably as will a short in the stator winding.
You can run a car without any alternator at all. The battery will provide the current to supply all the electrical load however it won't last long depending on how good your battery is and how much electrical load you have. Lights , Heaters, wipers, heated seats, heated windows, mirrors etc really stretch a charging system.
You may get away with just a new regulator on the alternator but probably better to replace it with a remanufactred item as sometimes they are a pain to change.