The sufficient provocation would be the guy you've been illegally pursuing suddenly lunging for your gun.
Their pursuit of Arbery was arguably reckless and undoubtedly illegal.
I don't think the pursuit, in and of itself, was illegal. The idea of following someone for the purpose of maintaining contact and relaying their location once officers arrive is surely legal. To then go on and initiate a confrontation is taking a much larger leap. I haven't run down everything but did find this.
GA 17-4-60
A private person may arrest an offender if the offense is committed in his presence or within his immediate knowledge. If the offense is a felony and the offender is escaping or attempting to escape, a private person may arrest him upon reasonable and probable grounds of suspicion.
As I type this I am not aware that either of the first criteria were actually met and have no reason to believe the assertion could be made a felony offense was in play. That could make it arguable that the entire pretense for the CA was questionable which is not good for anything that comes about after, which in this case is a death.