This reindeer can only be found during the Feast of Winter Veil. To tame it you need to have your gear transmogged to a full, four-piece 'Festive Holiday' transmog set, including the coat, pants, boots, belt (either Red or Green - but not a mix; and the vest versions don't seem to work currently, but the shorts do). These items can be purchased from Wulmort Jinglepocket or Penney Copperpinch near Greatfather Winter in Ironforge and Orgrimmar, respectively. Alternatively, you can also use the 'Disposable Winter Veil Suits' toy, though currently the green appearance doesn't seem to work, just the red one (but you can keep swapping). If you're not using the full appearance, the reindeer cannot be seen. Inside the dungeon, the solitary reindeer can be found atop the large spur to the east of the entrance, sleeping in a tree hollow at the southern end (around 36.0,39.0). The red-nosed version is much rarer, so you may need to reset the dungeon a number of times to find one but you can '/tar dreaming festive reindeer' from a short way into the entrance tunnel (a bit past the first bend) and its portrait will reveal which appearance it has (but if you enter the instance before donning the transmog set it won't appear until you reach its location!). Much of this information comes from Paryah and PigeonD at Wowhead.
Level scaling: Most NPCs will scale with the Hunter's level, within the constraints of their level range. Hunters can tame regular NPCs up to 2 levels higher than them, but can only tame elite NPCs of the hunter's level or below.