On Wikipedia I found the Latin text of the Exsultet, but it is not clear when exactly it was written, or by whom. What is its origin?
According to Zenit and the Catholic Encyclopedia, the solemn rite originated no later than the late fourth century, with the current text of the Exultet likely dating back to the fifth century. The earliest extant manuscript is found in the Bobbio Missal, dating from the seventh century. Its author is unknown.
The Encyclopedia of Ancient Christianity says:
The Exultet’s origin is uncertain, but it is clear that it was inspired by Ambrose. Textual analysis shows, if not Ambrose’s hand, at least his mind. [...] The part of the Exultet that speaks of bees was inspired by Virgil. [...] In his letter to Praesidius, Jerome derides this praise of the bees as entirely out of place in the Easter liturgy.
From the end of the 4th c. the singing of the praeconium paschale was the task of deacons.
If written by Ambrose, it had to have been before his death in AD 397. The current text may be different from the earliest versions, but Jerome's letter regarding the bees would have been written before his death in 420, confirming that at least that content was in the text by then.