It was not lawful for the Sanhedrin to put Jesus to death

Herod beheaded John in Mt 14:10 without a Roman trial.

They tried to stone Jesus in Jn 8:59 outside of the Roman law.

They successfully stoned Stephen to death in Ac 7:58 outside of the Roman law.

Some Jews in Damascus conspired to kill Paul in Ac 9:23. Some years later, the chief priests and the elders conspired with 40 assassins to kill Paul in Ac 23:15.

These were some examples of executions outside of the Roman legal system.

Jn 18:

28 They led Jesus from the house of Caiaphas to the governor’s headquarters.f It was early morning. They themselves did not enter the governor’s headquarters, so that they would not be defiled, but could eat the Passover. 29 So Pilate went outside to them and said, “What accusation do you bring against this man?”

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The Sanhedrin delivered Jesus to Pilate instead of executing him themselves.

30 They answered him, “If this man were not doing evil, we would not have delivered him over to you.” 31 Pilate said to them, “Take him yourselves and judge him by your own law.” The Jews said to him, “It is not lawful for us to put anyone to death.”

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Sure, but then why did the Sanhedrin go through the trouble with Pilate? Why not just assassinate Jesus secretly?

Pilate was their hired gun. They wanted to use Roman law to kill Jesus. The fact that it was not lawful for them to put Jesus to death was not their worry. Their real worry was the repercussions from the populace if they had assassinated Jesus. There were Jews at large who were sympathetic to Jesus, particularly during the Passover Festival. On Palm Sunday, Lk 19:

35 They brought [the colt] to Jesus, and throwing their cloaks on the colt, they set Jesus on it. 36 And as he rode along, they spread their cloaks on the road. 37As he was drawing near—already on the way down the Mount of Olives—the whole multitude of his disciples began to rejoice and praise God with a loud voice for all the mighty works that they had seen, 38 saying, “Blessed is the King who comes in the name of the Lord! Peace in heaven and glory in the highest!” 39 And some of the Pharisees in the crowd said to him, “Teacher, rebuke your disciples.” 40 He answered, “I tell you, if these were silent, the very stones would cry out.”

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The smart thing for the Sanhedrin to do was to 'hire' Pilate to kill Jesus for them.

Did the Sanhedrin really worry about the fact that it was illegal, according to Roman law, for them to execute Jesus?

No, not really.

Why didn't the Sanhedrin hire assassins to kill Jesus?

Jesus was a high-profile and popular figure. They worried about the repercussions. They thought they could get the people to blame Pilate and Rome, but Pilate washed his hands (Mt 27:24).