The 'legion' was abandoned long before the split into East and West and the development of the Eastern Empire into what we now call the Byzantine Empire.

The stereotypical Roman legions in their metal plate lorica segmentata only existed in a very specific timeframe of early Imperial Rome. Here's a nice image I just googled showing a bit of the evolution of Roman soldiers.

If you're asking when the Byzantines started using mercenaries. I'm not sure what you're referring to, nor am I a particular expert on that period, but afaik the shattering of the Empire following the loss of Constantinople to the fourth crusade in 1204 is the most decisive moment from where the Empire became increasingly reliant on various forms of outside support. (And personally, but you need not agree with this, I consider it a point of discontinuity, with the Palaiologoi Empire a successor state, not a continuation of the Byzantine Empire.)

The Roman army had abandoned the traditional Legion long before the separation, as has been pointed out already. The later Roman army was organised into Comitatenses, varying units of Legio Palatina and Auxilia Palatina, as the infantry components of their field armies, with the rest of the infantry, essentially the rumps of the legions as garrisons of their depots and fortresses. These were called Limitani or Ripenses, and, over time, became regarded very much as static second line troops, who would only occasionally be used as part of a field army, referred to then as Pseudocomitatenses. Most of the Palatina units were newly raised, either entirely from scratch, or from existing Legios and Auxiliary Cohortes, replacing the ad hoc useage of detachments from Legios and Cohortes, called Vexillationes. There is a very comprehensive list of all of the units of the Roman Army in a document called the Notitia Dignitatum. From that one can trace the Eastern Roman units that became the basis for the Byzantine Army.