Horses and donkeys have never been shown in a Fallout game, but have been mentioned on occasion. Some equidae appear to have survived the Great War and only lived for a short time, before becoming extinct in just under or over a century onwards. One such case is Pugsley, a donkey owned by a man named Irwin who was killed when his ranch was attacked by raiders around 2161. The Chosen One when talking to Ethyl Wright however mentions abundance of mare's milk being around among tribesmen.
In All Roads, the graphic novel, every NCR soldier seen in Chance's flashback of the Bitter Springs Massacre is riding a horse. Given the number of continuity errors between All Roads and Fallout: New Vegas it is likely this is merely another such occurrence.
When asked about the horses seen in All Roads, Chris Avellone said, "To me, that picture in the comic was an error (no one noticed it until too late) and that "it was never my intention that there were any horses." He went on to say that "there were parts of the New Vegas design where we specifically removed the mention of horses from certain narrative points (one of Raul’s stories originally referenced horses, for example, and we cut them because we didn’t want to imply horses still existed)."
In the Fallout Bibles, Chris Avellone refers to Pugsley as a mule, rather than a donkey. A mule is a cross between a donkey and a horse, which would imply both survived; however given that Irwin in game uses the term donkey, this is likely just an error on his part.