Yes, they are on skis. The domesticated reindeer was often raised from a very yound age among humans, so possibly some of them were domesticated enough to carry a man. The source doesn't appear to be fanciful. However, we know of latter Finnish cavalry using horses, so maybe the reindeer was available, but not the best mount. There has been a recent report in German news about a girl riding a cow (http://www.nordbayern.de/nuernberger-nachrichten/region-bayern/sie-wollte-ein-pferd-jetzt-reitet-sie-auf-einer-kuh-1.110086
and riding oxen in races is a kind of fun entertainment (http://www.ochsenrennen.de/). So I think it perfectly possible to ride the strangest animals (http://www.creepysantaphotos.com/2009/12/07/santa-riding-an-ostrich/).