This is my first reply post on r/AskHistorians but I think I can help out as this is an area I've been studying recently. The short answer is both Yes and No. A few things to begin with, while 0 A.D. is traditionally thought to be when Jesus was born, it is now thought to be closer to 3 A.D. to 6 A.D. because of the reigns of the rulers mentioned. This is because in 6 A.D. there was a census conducted by Publius Sulpicius Quirinius who was governor at the time. The other thing to keep in mind is that there are a LOT of Herods. There is Herod Agrippa, Herod the great, and Herod Archelaus.
The other thing is that throughout history, the Jews and the Romans did not get along at all. Especially around Passover, it was common for revolts and uprisings to happen in Jerusalem. Therefore, the Romans would allow for a ruler of Jewish descent to rule over Judea as long as that ruler was beholden to Rome.
First, you have Herod the Great who was the kind of Judea. While he was king, this power was given to him by the Roman Senate and he, interestingly enough, had a very close and personal relationship with Marc Antony.
From Josephus' The Wars of the Jews Chapter 14:
Hereupon Antony was moved to compassion at the change that had beenmade in Herod's affairs, and this both upon his calling to mind how hospitablyhe had been treated by Antipater, but more especially on account of Herod'sown virtue; so he then resolved to get him made king of the Jews, whomhe had himself formerly made tetrarch. The contest also that he had withAntigonus was another inducement, and that of no less weight than the greatregard he had for Herod; for he looked upon Antigonus as a seditious person,and an enemy of the Romans; and as for Caesar, Herod found him better preparedthan Antony, as remembering very fresh the wars he had gone through togetherwith his father, the hospitable treatment he had met with from him, andthe entire good-will he had showed to him; besides the activity which hesaw in Herod himself. So he called the senate together, wherein Messalas,and after him Atratinus, produced Herod before them, and gave a full accountof the merits of his father, and his own good-will to the Romans. At thesame time they demonstrated that Antigonus was their enemy, not only becausehe soon quarreled with them, but because he now overlooked the Romans,and took the government by the means of the Parthians. These reasons greatlymoved the senate; at which juncture Antony came in, and told them thatit was for their advantage in the Parthian war that Herod should be king;so they all gave their votes for it. And when the senate was separated,Antony and Caesar went out, with Herod between them; while the consul andthe rest of the magistrates went before them, in order to offer sacrifices,and to lay the decree in the Capitol. Antony also made a feast for Herodon the first day of his reign
Herod the Great died of disease around 4 A.D. In his will, he created a Tetrarchy, which split his kingdom among his sons in 1/4 shares, though not 1/4 equal shares. The two that are most relevant here are, of course, also named Herod. Herod Archelaus received the largest portion of the kingdom which amounted to 1/2 of the kingdom whereas Herod Antipas received Peres and Galilee. These amounts and areas are described in Chapter 11 of Josephus' Antiquities in book 17.
Josephus also states that Archelaus was appointed the "ethnarch" and that Caesar would give him the "royal dignity" if he "governed his part virtuously". This means that while the Herods were subject to Rome, they also ruled their respective areas. Further, Archelaus would be considered the king of the ethnic Jewish people as per his title. However, in the 10 year of his reign, Archelaus would be exiled to Gaul for not "governing virtuously"
From Josephus' Antiquities Book 17 Chapter 13:
But in the tenth year of Archelaus's government, both his brethren,and the principal men of Judea and Samaria, not being able to bear hisbarbarous and tyrannical usage of them, accused him before Caesar, andthat especially because they knew he had broken the commands of Caesar,which obliged him to behave himself with moderation among them. WhereuponCaesar, when he heard it, was very angry, and called for Archelaus's steward,who took care of his affairs at Rome, and whose name was Archelaus also;and thinking it beneath him to write to Archelaus, he bid him sail awayas soon as possible, and bring him to us: so the man made haste in hisvoyage, and when he came into Judea, he found Archelaus feasting with hisfriends; so he told him what Caesar had sent him about, and hastened himaway. And when he was come [to Rome], Caesar, upon hearing what certainaccusers of his had to say, and what reply he could make, both banishedhim, and appointed Vienna, a city of Gaul, to be the place of his habitation,and took his money away from him.
This shows that, while the "king" had some power within his territory, it was not independent or Rome and Rome was more than willing to step in, with harsh consequences.
The Herod when Jesus was born was Herod Antipas who ruled over Galilee. While Rome could have stepped in, as stated before, Rome only stepped in when necessary. While the legal process that sent Jesus back and forth from Pontius Pilate and Herod Antipas is debated and something I'm not familiar enough with to discuss at length, the answer to your question is that, has Herod Antipas wished for Jesus to go free, he would have had that authority as Antipas ruled over Galilee where Jesus was from.
So in short, to answer your question, Judea did have kings but only one who held the title of "king" and he was later exiled. Further, that title was mostly symbolic and referred to the power he had over his people rather than in relation to Rome. As, as illustrated by the accounts of the Herods as told by Josephus, they knew that they had some power over their territory, but also knew Rome could step in at any time, with serious consequences. Also, due to the Jewish disdain of Rome in general, the Herods had to walk a fine line between keeping Rome happy and keeping their people happy. While not strictly a puppet, as they had power to rule over their own domains, there were strings attached, no pun intended. So in short, it's best to think of the Herod tetrarchy as foreign backed rulers than as king or puppets. Hope this helped!
Edit: Formatting
Edit 2: Spelling/Formatting