Nah, he has a ship (steamboat) and a horse called Americo. And if memory serves me right, the English version is derived from the American version which was derived from the Dutch Sinterklaas...
No Reindeer!!! "Boo!" Animals have employment rights too! Is that why he has that laaarge red Superman cape, to fly around the world instead of being dragged by Donner, Blitzen, Rudolph et al? :D