They did use archers, from at least the Late Republic and into the Imperial Period, but our information about them is scattered and without good detail. They were never "Roman" units, always auxiliary, and were frequently drawn from Crete (the "Cretan Archers" were a famous group) or, later, Palmyra. Their role on the battlefield in all periods is unclear, and like archers from all periods, they could not stand up to infantry. We hear about them several times being used on ships in naval battles. Some examples:

In Caesar's first invasion of Britain in 55 BCE, he uses archers on board ships to aid the landing on the beaches at Deal:

When Caesar observed this, he ordered the long ships, the appearance of which was rather strange to the barbarians and the motion more useful for the occasion, to be withdrawn a little from the transport vessels, and to be propelled by their oars, and be stationed on the open flank of the enemy, and the enemy to be pushed back and driven away by slings, arrows, and catapults: which plan was of great service to our men; for the barbarians, being terrified by the appearance of our ships and the motions of our oars and the nature of our catapults, which was strange to them, stopped, and even after retreated a little. (BG 4.25)

Famously, M. Antonius included archers on his ships at the battle of Actium vs Octavian, even though it seems that both of the generals had anticipated a confrontation on land rather than at sea. It is unclear if Antony intended the archers to be used at sea, or in an subsequent land battle.

Tacitus, in his description of Germanicus' campaign against the Cherusci (Ann. 2.9-17), talks about archers (sagittariī) in the Roman ranks as well, both on foot and mounted(!). The infantry of Arminius threatens the archers and they must be saved by other Roman forces. We also hear about archers under Poppaeus Sabinus against the Thracian rebels (Tacitus Ann. 4.47):

Then, after fortifying a camp on the spot, Sabinus with a strong detachment made himself master of a narrow mountain-ridge running without a break to the nearest tribal fortress, which was held by a considerable force of armed men and irregulars. Simultaneously, he sent a picked body of archers to deal with the bolder spirits who, true to the national custom, were gambolling with songs and war-dances in front of the rampart. The bowmen, so long as they operated at long range, inflicted many wounds with impunity; on advancing closer, they were thrown into disorder by an unlooked-for sally, and fell back on the support of a Sugambrian cohort, drawn up a short distance away by the Roman general, since the men were prompt in danger, and, as regards the din produced by their songs and weapons, not less awe-inspiring than the enemy.

Here they certainly seem to operate very much like archers in medieval battles, though they did not have the famous longbows of the later period.

Later on in the Imperial period, we start to see archers integrated more into the army as it transitioned from a Caesarian/Augustan structure to a late antique structure. In the late period we even hear about a whole legion (smaller than an Augustan legion), legio I Isaura sagittaria (Notitia Or. 7.56).

A good source for this question (but not the late period) is: Brudenell, F. 2002. “Fundis sagittisque: Roman military archery from the republic to the second century ad,” in P. R. Hill (ed.), Polybius to Vegetius: Essays on the Roman Army and Hadrian’s Wall Presented to Brian Dobson. Oxford.