The carroballista is attested throughout the second century AD. It appears both on Trajan's Column, which depicts the Dacian War at the start of the century, and on the column of Marcus Aurelius constructed c.190AD. It seems to have gradually fallen out of use after that.

The carroballista was most likely developed at the end of the first century in the reign of Nerva. During this time, Roman artillery technology and tactics were being overhauled. We don't know why - even in Nerva's reign the general Frontinus wrote in his Strategemata that artillery had 'long since reached its limit' and he saw 'no further hope' for tactical development. But even as he wrote this, engineers in Alexandria were devising a revolutionary new range of ballista designs, which hit the battlefield during Trajan's Dacian War in 101 AD. We know from archeological finds, in particular the dimensions of the holes which torsion spring rope passed through, that there were a wide variety of sizes from small handheld machines to heavy duty, 1.25m wide pieces. They were much more poweful than the previous generation of artillery (which was hundreds of years old by this point), and had a modular design where parts could be swapped out in the event of failure. One of the problems of the old design was that if one of the torsion springs failed the entire machine would have to be taken apart. Now they just unclipped the broken spring and put a new one in. Trajan's Column even depicts artillerymen carrying spares on their belts. This was a massive overhaul of technology and tactics.

This is where the carriage ballista comes in. Truly mobile artillery had never been a thing, as speed of deployment was limited by a man's ability to carry a heavy piece of equipment over the terrain. In theory, the carriage ballista could move as fast as a chariot, fire of a round, then retreat to safer places. Reconstructions and mathematical modelling have found that it could comfortably fire a bolt over 500m. They could, in theory, act as long distance artillery, close support for infantry, or do drive by attacks on the flanks. They were, in theory, a very innovative and versatile tool for any Roman general.

We don't know much about how they were actually used. Sources for the second century AD on military machinery in general are very poor. On Trajan's Column, the carroballista is shown following closely behind the infantry as they advance, which suggests that they were used to support the century (the Roman fighting unit of 80 men) that they were assigned to. According to the late Roman writer Vegetius, each century had a carroballista that was supposed to follow them around. Of course, a general could detach them and have them fight independent of the infantry if they wished. We don't really know how they were used other than as close infantry support, and even that we only know from Trajan's Column (the column is generally keen to show off the army's new tech so it features pretty heavily). Trajan certainly liked them, and they were useful enough to be used throughout the century, but we don't really know how widely they were used or how they were deployed in battle.

As for why it died out, we can be fairly confident that the carroballista was not cost effective. It had the combined advantages of artillery and chariots, but most of the drawbacks too. The operators needed training in both chariot driving and artillery operation, which would have made them some of the most specialised men in the entire army, and a nightmare to replace when they died or retired. The army had a problem retaining enough artillerymen as it was without the addition of this extra specialised unit. In the later writings of Ammianus Marcellinus in the 4th century AD, we see an increased focus on versatility and transferable skills within the army (or at least Ammianus thinks that's what they should do), which might have led to the hyper-specialised training becoming unfavourable. Chariots are also expensive to maintain, take a long time to replace losses (due to that specialised training), and need an open battlefield with few natural obsticales to be most effective. In the 3rd century AD, the Roman economy collapsed and the army needed to save money on equipment. They ditched most of their fanciest equipment in this time; the segmented armour fell out of use, as did the carroballista. It's also worth noting that the carroballista's little cousin, the manuballista, survived and thrived during the 3rd century. This was a handheld version that one person could use, kind of like a heavy crossbow. It was small and light enough that it could move as fast as the operator could run without sacrificing much power, which limited the extra value of mounting a ballista on a chariot for mobility's sake.

So they were used in the second century, but were probably not worth the financial and training investment when the army had to save money during the economic crisis of the following century, especially when the handheld manuballista was still an option.