I've actually questioned this myself in the past, here's what I've basically found so far
After digging for a while on Japanese Q&A sites and forums, most people seemed to have concluded that it's referring to a "Paddle", as in an implement to whip the reindeer with.
However like the Western community, they have about as much of an idea as we do on what it's actually supposed to mean as I don't think there's even been an official comment on its meaning, there's even a difference between how people think it's supposed to be written, as パドル or パドる, the difference being the later is a verb and the former being just a noun.
Either way, this song is completely made up, it's just for TYPE-MOON ACE, so it could just as easily be utter nonsense gibberish to fill in the tune.
To add more context, here's a thread about this: http://フェイト-fate-grandorder攻略速報.com/lite/archives/50387834/comments/5642764/
The OP opens up asking what the Padoru (パドル) part means and some correct him that it's normally パドる, some say otherwise, however as far as I can see, パドる is more a widely accepted way of writing it, with pixiv dic and many YouTube videos seeming to have this spelling.
Post #5 on the thread sums it up kinda well:
It's not the noun "Paddle", it's the verb "Paddle", as in the act of Paddling, or at least that's how I interpreted it.