The Lamut/Even and neighboring peoples of Siberia utilized reindeer as cavalry mounts though I could find only one paper providing details: Some Aspects of Warfare of the Reindeer Lamut in the 17th century by Alexander Nefedkin. Nefedkin cites Cossack eyewitness who states that the Lamut fight "sitting on reindeers as if riding on horses" and other accounts which state that the Tungus people (a neighbor to the Lamuts/Even) were trained to shoot bows from the back from the reindeers; the Lamuts/Even were also reported to shoot game from the back of reindeers.

There are no sources indicating the specific tactics of the reindeer cavalry though due to the emphasis of these cultures on archery, Nefedkin posits that the reindeer cavalry might be primarily mounted archers.

Follow-up question if anyone's still checking here: I heard the Swedes experimented with moose-born cavalry in the 17th century but couldn't get the moose to shy from pikes, leading to a lot of unnecessary moose deaths. Is this true?

Can somebody verify this portion of Wikipedia, which is almost entirely source-free? As the boxing announcer in Futurama says, "Interesting if true!"

Ancient Egyptians often kept cheetahs as pets, and also tamed and trained them for hunting, although they did not domesticate it. Cheetahs would be taken to hunting fields in low-sided carts or by horseback, hooded and blindfolded, and kept on leashes while dogs flushed out their prey. When the prey was near enough, the cheetahs would be released and their blindfolds removed. This tradition was passed on to the ancient Persians and brought to India, where the practice was continued by Indian princes into the twentieth century. Cheetahs continued to be associated with royalty and elegance, their use as pets spreading just as their hunting skills were. Other such princes and kings kept them as pets, including Genghis Khan and Charlemagne, who boasted of having kept cheetahs within their palace grounds. Akbar the Great, ruler of the Mughal Empire from 1556 to 1605, kept as many as 1,000 cheetahs. As recently as the 1930s, the Emperor of Ethiopia, Haile Selassie, was often photographed leading a cheetah by a leash.

This may sound nitpicky, but there weren't just Llamas in South America, there were Llamas, Alpacas and the Chilihueque which may or may not had been a different species.

No historical accounts of Llama riders, though it's technically possible they are very temperamental and nasty, so I wouldn't be surprised if nobody bothered. There's an account from a dutch navigator seeing them used for ploughs though.

The Roman author Aelian noted that pigs were sometimes used by Romans and Greeks to break sieges by war elephants, since apparently, elephants were scared by pigs, which would cause great confusion and sometimes make the elephants turn around and stampede into the enemy's own ranks.

Of course they wouldn't ride the pigs, but it's still an interesting use of an animal for warfare.

(source: Aelian, On Animals 1.38; 16.36)

If you want a fun read related to the subject, you might wanna pick up Greek Fire, Poison Arrows, and Scorpion Bombs: Biological and Chemical Warfare in the Ancient World by Adrienne Mayor.