The hypothesis has existed since the days of Early Christianity; it is discussed in the Gospel of Matthew, generally agreed to have been written between AD 70 and 100. Matthew's gospel raises the hypothesis only to refute it; according to it, the claim the body was stolen is a lie spread by the Jewish high priests.
The primary sources of details about Jesus are the Gospels. Roman records are spottier - there is no extant contemporary record of the execution of Jesus, for example, not that such a thing would be expected, and thus no details about what was done with the body afterward. As such, accounts of the days between Jesus's execution and the discovery of the empty tomb are almost exclusively based on the Gospel accounts and knowledge of society at the time, and it is difficult to say more than scenarios such as the stolen body hypothesis are "plausible" or "unlikely," rather than "proven" or "disproven".
According to the Gospel of Mark, generally thought to be the oldest of the gospels, Joseph of Arimathea asked Pontius Pilate for the body of Jesus. Afterward, a group of women went to the tomb, and found the stone rolled away, an angel there, and no body. The Gospel of Luke largely concurs with this account, though the list of women slightly differs. According to the Gospel of John, Nicodemus helped Joseph of Arimathea with the burial of Jesus. It also notes that Jesus was buried in a garden near the site of the crucifixion, and that no body had lain there before. In John, Simon Peter and the beloved disciple also come to the tomb to verify Mary Magdalene's claim of an empty tomb; there is no direct reference to this in Mark and Luke, where it is implied that the apostles only believe upon seeing the resurrected Jesus.
The Gospel of Matthew includes a distinct account of the period between Jesus's death and the discovery of the empty tomb not in the other gospels, and directly addresses skepticism about the resurrection. In Matthew's account, the chief priests and the Pharisees know of prophecies that Jesus will return in three days, and fear that his disciples will steal the body to make it appear that he has been resurrected. They ask Pilate to secure the tomb, and Pilate sends a guard to watch the tomb. When Mary Magdalene arrives at the tomb, unlike the accounts in the other gospels, there is an earthquake and the tomb rolls open in front of her. An angel appears and scares away the guards, and the empty tomb is revealed. When the guards report this to the chief priests, the priests bribe the guards to lie about the events:
...some of the guard went into the city and told the chief priests everything that had happened. After the priests had assembled with the elders, they devised a plan to give a large sum of money to the soldiers, telling them, "You must say, 'His disciples came by night and stole him away while we were asleep.' If this comes to the governor's ears, we will satisfy him and keep you out of trouble." So they took the money and did as they were directed. And this story is still told among the Jews to this day.