The verse in question gives rise to two considerations - first, what it actually does say about those persecuted Christians and, second, what it does not say.
What it does say:
When in Jerusalem, Saul of Tarsus imprisoned many Christians, having been given that authority from the chief priests.
Saul of Tarsus gave his vote towards having those ones put to death.
What it does not say:
It does not say that Saul of Tarsus personally put them to death. (We know he held the garments of those who stoned Stephen to death - Acts 8:1, or should that be Acts 7:60?)
It does not say that the killing of those Christians was legal. If the Jewish religious leaders here acted illegally, they would have done it behind the Romans' backs, while they weren't looking, when they expected they could get off with that crime. That could be indicated in the case of the woman caught in the act of adultery - John 8:1-11 - especially with a woman. Who knows but that if the adulterous man had also been captured alongside her, they would have felt obliged to get Roman permission first?
Conclusion: This question about Acts 26:10 can only be answered by consideration of the actual information and sticking only to what we are told. There is not enough information in the verse to assume that the Jews were allowed to carry out the death penalty at that time, which was after the time of Christ on earth. Those who conclude that this verse is a contradiction, or error, in the Bible, have jumped to a conclusion that is simply not warranted.