The question is more difficult to answer than it appears: modern scholarship on the army of the Roman Republic is considerably more scarce than on the army of the Principate, particularly when it comes to technical aspects of organisation and equipment rather than bigger-picture examinations of manpower or campaigns. The Republican army was not a professional force, and did not leave the kinds of records and evidence the permanently garrisoned legions and auxilia of the later empire did. The written sources we do have focus on big-picture stuff, and when they mention specific troops they very rarely talk about the light infantry, who would be the ones to carry missile weapons such as bows, and particularly unlikely to talk much about the light infantry of the Italian allies or other non-Roman forces.

It's fairly safe to say, though, that the short answer to your question is: "Not very." Archery didn't play a particularly big role in western Mediterranean warfare around that time.

Where the Persians held (according to Herodotus) that the most important things for man to know were to tell the truth, ride a horse, and shoot a bow, and where steppe peoples such as the Scythians or Parthians practised mounted archery much as they always did, and even Greek armies fielded light infantry armed with bows, neither the Romans nor the other people of the western Mediterranean seem to have made much use of archers.

Instead, the main missile weapon used by the Romans in Republican times was the javelin. The light infantry velites and/or rorarii carried bundles of light javelins, most legionaries carried the heavy pilum, and light cavalry such as the famous Numidians would likewise be throwing javelins rather than shoot bows. Some individual men may well have had bows and arrows, but if so our sources do not mention them.

That does not means archers were not used at all in Roman armies. Our sources do specifically talk about them at several points. However, in these cases, we are dealing with specialised mercenary contingents, usually the famous Cretan archers or Rhodian or Balearic slingers. (Digression: u/Iphikrates points out in this excellent thread we may be dealing with an issue of branding rather than nationality here: "Cretan Archer" can be read as a shorthand for "mercenary who's a good archer.")

These specialists are first mentioned by Roman historians as serving their enemies: Pyrrhus had Cretan archers with him, Hannibal employed the Balearic slingers to great effect. However, as time goes on and we move from the middle republic to the late in the 1st century B.C., Roman and Italian forces stop providing their own native light infantry (and cavalry) and the Romans become more reliant on foreign allies to provide missile troops. This process starts after the second Punic War, and becomes common practice until the fall of the Republic. Caesar had Cretan Archers with him in Gaul, (though there's no mention of them after 57 B.C.) and later in the civil war against Pompey, he mentions large levies of Greek and near-eastern archers and slingers serving Pompey's forces.

In Imperial times, the Romans cease to rely on mercenaries, and instead form their own professional auxiliary units, some of which were specialised in archery. These still tended to originate in the the eastern provinces, though, and Kepie states that these were the only units where we see new soldiers being recruited from their original home region.

So all in all, the conclusion is that archery was less of a feature in Roman warfare than even in Greek warfare, and that there is no evidence of native Roman archers in Republican times, although we must always keep in mind that this evidence isn't particularly exhaustive.

Sources:

Keppie, Lawrence, The Making of the Roman Army, Taylor and Francis, 1998 (but first published in 1984. Like I said, there's not as much recent material on the Republican army as I might wish.)

Goldsworthy, Adrian, The Complete Roman Army, Thames & Hudson, 2003

Goldsworthy, Adrian, The Punic Wars, Cassel, 2000