The crossbow was not commonly used by the Romans during Late Antiquity, and was fundamentally different than Medieval crossbows. The Late Roman crossbow was known as a manuballista. We have only fragmentary evidence for its construction (and almost none for its employment), though, as its name implies (manuballista translating roughly to “hand-ballista”), the manuballista was a version of the traditional ballista, small enough to be wielded by one man. As such, the weapon relied upon twin torsion springs to provide the energy necessary to propel a shaft. This distinguishes it from Medieval crossbows, which relied upon a bow shaft (known as a lath) pulled taught. A manuballista is thus not a crossbow in the traditional sense, as it relies upon torsion springs rather than a lath. My knowledge of Medieval crossbow construction is unfortunately limited, so I can't really say more, but it's worth pointing out that the manuballista is not technically a crossbow.
The manuballista appears to have been an uncommon weapon in Late Antiquity, as we have extremely few textual references to it. The composite bow was the common Roman missile weapon, carried by specialist regiments of archers and by the contingent of archers that accompanied most infantry regiments. Individual infantrymen typically carried multiple short darts (known as plumbatae) attached to the back of the shield, or several javelins. While the overall volume of missile fire clearly increased in Late Antiquity, the role of the manuballista remains unclear, though it was doubtless ephemeral at best. Given the lack of sources regarding its use, it is unsurpising there are none that address why it appears to have fallen out of use in Western Europe by the end of the fifth century.
Unfortunately there is no definitive answer to how widespread the use of the crossbow was in the Roman Empire. The 'crossbow' itself was invented many centuries before in Greece known as the gastraphetes, however this was most likely used only in sieges.
There are many depictions of handheld ballista-type weapons being used, but not depicted in battle or with little context at all. The term Cheiroballistra or manuballista is used in the late Roman Empire to describe such weapons, and it seems that they were an evolution of the earlier scorpio. There is currently no consensus on who developed the weapon, but many historians point toward Hero of Alexandria. The device seems to be comprised mostly of metal according to modern reconstructions, shot metal bolts and is smaller than the scorpio. Modern reconstructions also indicate that it was a two-man anti-personnel weapon, used to fire darts over range with reasonable accuracy. However, the term Cheiroballistra is a combination of the Greek words 'hand' and 'shooter', which implies that a handheld version may have existed.
The Notitia Dignitatum (list of offices) records a number of units active in the Roman army under the title of 'ballistarii'. This translates as 'artillery', however the large number of such units recorded in the Notitia would indicate that the term may likely have referred to crossbow armed foot soldiers (Cheiroballistra), rather than siege trains. Now, whether these were hand-held crossbows or teams of two or three we do not know.
To answer your question then - there is no historical consensus as to whether or not the crossbow we know today was used in the Late Roman Army. What fragmentary evidence we have on the existence of the cheiroballistra tells us nothing of what it was used for or if it even saw widespread use. How exactly handheld crossbows seen depicted were used in Roman times is an ongoing debate - whether they were used in war, for hunting and how widespread their use was. Personally I would argue that handheld crossbows were likely used to some extent, however probably very little, due to the various depictions of crossbows such as here and the term 'hand shooter' translated from the Greek/Latin word used to describe Hero's invention. When the weapon was used at all, it was most likely a weapon deployed in sieges where the slow reload time and other disadvantages compared to a bow/sling were not as important.
A similar question that might help
Specifically the top answer by /u/SmilesFTW, which addresses the pros/cons of the crossbow versus the longbow
Related question (forgive me if this is not allowed): when did it come BACK into popularity, for use mostly in hunting? Why does my local pawn shop sell them (and bows) when the fun dominated for so long ?
