Where is there official writings from Perpetua or Felicity? Supposedly there was a diary written by a woman named "Perpetua" just before she was killed in the arenas for her faith. 

Another similar account is concerning Felicity. 

Did these women actually write down anything authentic, and if so, where are their writings found?

I've found plenty of commentaries and history accounts, but no actual writing from them.

Do they have written accounts and where can they be found?

Such a thing would show the mind of a Christian during that time. Their views, thoughts, motives, etc, possibly distinct from modern Christians.
This site provides a copy of the purported prison diary written by Saint Perpetua, written around 203 CE. The diary forms part of the non-canonical Acts of Perpetua and Felicitas, also known as The Passion of St. Perpetua, St. Felicitas, and their Companions. It is a beautifully crafted story and many believe Perpetua and Felicity really existed and that Perpetua's diary is authentic, although it has been edited. Assuming its authenticity, Perpetua's original diary no longer exists.

There are two saints known as Felicity.  Saint Felicity (Felicitas) is a legendary martyr said to have lived in Rome between about 101 and 165 CE. This information comes from a sixth-century 'Acts', which may have been the first record of St. Felicity, as her name does not appear in the well-known fourth-century Roman calendar. Another legendary Saint Felicity was said to have been a slave of Perpetua, and both died as martyrs in Carthage in 203 CE. 

In Ancient Fiction: The Matrix of Early Christian and Jewish Narrative (edited by Brant, Hedrick and Shea), pages 249-250, Dennis R. MacDonald discusses parallels between the the deaths of Perpetua and of Polyxena, which is narrated in Euripides' ancient story of Hecuba.  Thomas J. Heffernan, in The Passion of Perpetua and Felicity, page x, acknowledges there is much in the depiction of Perpetua that owes to earlier texts, and that it is purposely intended to parallel the heroism of Perpetua with that of Polyxena, but concludes there is a historical core to the story.