The earliest accounts are Ammianus Marcellinus (4th century AD) and Vegetius (end of 4th c. and beginning of the 5th c.).
According to E. W. Marsden: "To sum up the situation in the first three imperial centuries, the Roman army reached very high standards in artillery equipment and organization. In particular, after the introduction of the arrow-shooting ballista in its various forms at the beginning of the second century, and while the stone-throwing ballista remained in service, every Roman legion possessed substantial batteries o f the most powerful artillery produced in the ancient world. There is very little direct evidence for the third century. But the introduction of the onager into general service, which may have occurred in the second century, and of the non-torsion arcuballista, both machines being of relatively simple construction, suggests a shortage of good artificers."
The hypothesis is that Vegetius is using a lost text of Publius Tarrutienus (2nd century AD). There is also in a fragmentary text by Apollodorus of Damascus a mention of a one-armed machine and a variant of it incorporated to a battering ram. Furthermore, Apollodorus of Damascus was an engineer close to Trajan.
Previously, Philo of Byzantium, who lived most of his life in Egypt, under the Ptolemies around 220 BC, mentioned one-armed devices in the list of machines that could be used by the defenders against besieging platforms. No description survived.
I would say there is no evidence for its usage during the Roman republic and if we decide that Philo of Byzantium indeed mentioned onagers, then the Greeks were the first to use them.