Did the Jewish leaders have a law to put Jesus to death, or was it not lawful for them?

The two statements are about two different kinds of law, and John reports both within a short time without any sense of strain. In John 19:7 the leaders appeal to the law of Moses. In their eyes Jesus had blasphemed by claiming to be the Son of God, and that law says “he that blasphemeth the name of the Lord, he shall surely be put to death” (Leviticus 24:16). In John 18:31 they are speaking of what they were permitted to do under Roman rule. Judea had been governed by a Roman official since AD 6, and Josephus says the first of them was sent with the power of life and death; a tradition in the Talmud, which Gill cites, says that the Jewish courts stopped judging capital cases about forty years before the temple fell. The Sanhedrin could try a man and declare that he deserved death, as it did with Jesus (Mark 14:64), but carrying out the sentence needed the governor.

So both are true: by the law of Moses, as they read it, Jesus deserved to die; under Roman law they could not execute him. That is why they brought him to Pilate, and why they first accused him of a political crime, claiming to be a king (Luke 23:2), and only fell back on the religious charge when Pilate would not convict. John adds that this fulfilled Jesus’ own words about the manner of his death (John 18:32; 12:32–33): a Roman execution meant being lifted up on a cross, not stoning.

Scholars still discuss whether the Sanhedrin kept any power of execution in special cases, such as a Gentile entering the inner courts of the temple, and how to understand the stoning of Stephen (Acts 7:57–60), which may have been mob action. These questions do not affect the answer here.