Reindeer were a mammalian species of the deer family, often associated with Santa Claus.
History
Needs to be organised chronologically.
The Doctor, their companions and acquaintances occasionally encountered reindeer — and, indeed, Santa Claus himself. Once, Santa sent a sleigh pulled by reindeer to Dr. Who, John and Gillian to take them to his house. (COMIC: A Christmas Story [+]Bill Mevin, TVC comic stories (Polystyle Publications, Ltd., 1965-1966).)
On another occasion, the Third Doctor and Sarah Jane Smith met St Nick on a New York estate — an encounter which was witnessed by Clement C. Moore. (PROSE: A Visit from Saint Nicholas [+]Lisa Miles, Short Trips: Christmas Around the World (Short Trips short stories, Big Finish Productions, 2008).)
Every Christmas Eve, Lucie Miller laid out a trail of toast crumbs to her house, believing this would help Father Christmas's reindeer find their way. (AUDIO: Death in Blackpool)

At Hyperville — Europe's largest shopping mall in 2013 — the Tenth Doctor encountered the Snow Queen, who, like Santa, chose to travel in a sleigh pulled by reindeer. (PROSE: Autonomy [+]Daniel Blythe, BBC New Series Adventures (BBC Books, 2009).) Some time later, his TARDIS was pulled by reindeer after he discovered a group of them among the snow. (TV: The Doctor and the Reindeer [+]Russell T Davies, BBC idents (BBC One, 2009).)
In a dream, three reindeer, Rudolph, Donner and Blitzen, flew around Clara Oswald's home after Santa's sleigh crashed on the roof. Santa later rode into a polar base on Rudolph to stop the dream crab attack. Santa claimed to Shona McCullough that his flying reindeer were a "scientific impossibility", and as a result, he fed them "magic carrots" so they could fly. (TV: Last Christmas [+]Steven Moffat, Doctor Who Christmas Special 2014 (BBC One, 2014).)
According to one account, Santa's reindeer differed physically from normal reindeer, apparently being "a shade greener" in colour, and possessing at least two more legs. (COMIC: Frosty the Snowdemon [+]Christopher Cooper, DWA comic stories (BBC Magazines, 2008).)
Of course, not all encounters with reindeer were associated with Santa. The Eighth Doctor and Izzy Sinclair once met some Noxmycota-crazed reindeer when they visited Sweden in 1070. (PROSE: Illumination [+]David Bailey, Short Trips: Christmas Around the World (Short Trips short stories, Big Finish Productions, 2008).)
Cultural references to reindeer
Reindeer were so associated with Santa Claus that replicas were often made of them for Christmas decorations. The Tenth Doctor and Donna once landed at a "Santa's grotto" in a garden centre which contained large reindeer figures. (COMIC: Frosty the Snowdemon [+]Christopher Cooper, DWA comic stories (BBC Magazines, 2008).) Artists often included reindeer in their Christmas illustrations — just as Jasmine Deering did in her advertising campaign for Blackstone soft drinks. (PROSE: Christmas in Toronto [+]Andrew Cartmel, Short Trips: Christmas Around the World (Short Trips short stories, Big Finish Productions, 2008).)
Behind the scenes
Information from invalid sources

A Christmas stocking was decorated with an illustration of a reindeer. (TV: BBC One Christmas Film 2016 [+]BBC idents (BBC One, 2016).)