Yes, but only with explicit Roman approval.
The authority of the Sanhedrin
The Jewish Sanhedrin had the authority to try criminal cases, but could not (at the time of Jesus' death) carry out the death penalty (see John 18:31). This is why the Sanhedrin had to take Jesus to Pilate for condemnation (though they may well have had other ulterior motives as well, see here).
The Talmud may indicate that the Sanhedrin was stripped of the power of carrying out capital punishment in approx. AD 30 (see here), though the precise timing & meaning of the Talmud's statement is uncertain.
This didn't entirely put a stop to executions that weren't sanctioned by Rome. The people in Jerusalem used the leadership vacancy after the death of Porcius Festus in AD 62 to illegally put James (the Lord's brother) to death (see Josephus Antiquities 20.9).
Methods of execution
Crucifixion was the brutal, humiliating Roman method of execution (for non-citizens), but Talmage indicates that with Roman sanction the Jews could themselves had carried out execution by stoning:
had Pilate approved the death sentence and handed the Prisoner over to
the Jews for its infliction, Jesus would have been stoned (see Jesus
the Christ pp. 623-633)
The trap being laid in this passage
This passage is an example of a trap being laid for Jesus--the point of bringing the woman before Jesus was not that the hypocritical accusers cared what happened to her, but to try to get Jesus to say something inappropriate.
Assuming the reliability of the account, the trap was designed to work like this:
- If Jesus approved the stoning, He would be delivered to the Roman authorities for challenging their authority (since the Roman authorities had to approve the death penalty)
- If Jesus did not approve the stoning, He would be accused of heresy for teaching something contrary to what Moses had said
But Jesus outsmarted the trap and did not answer the question on their terms.