The guards at the tomb would need to be bribed.
The problem of bribing works both ways. The disciples could have bribed the guards to not say anything about them taking the body of Jesus. But, the Jews could also have bribed the guards into saying that the disciples stole the body of Jesus. In fact, the only bribing we see in the gospel accounts of the guards is done by the Jewish leaders. Matt. 28:11-15 says, "Now while they were on their way, behold, some of the guard came into the city and reported to the chief priests all that had happened. 12And when they had assembled with the elders and counseled together, they gave a large sum of money to the soldiers, 13and said, "You are to say, ‘His disciples came by night and stole Him away while we were asleep.’ 14"And if this should come to the governor’s ears, we will win him over and keep you out of trouble." 15And they took the money and did as they had been instructed; and this story was widely spread among the Jews, and is to this day."
The bribe would be very necessary since the guards were at risk of their lives if they had failed at their duty. It was the custom of the Roman military that if a prisoner escaped, then the guard(s) who was in charge of guarding the prisoner would take the prisoner's place. This is a very strong motivation to make sure that they carried out their duty, bribe or not. In Acts 16:25-30 when Paul was in prison, there was an earthquake that opened the cell doors which would have allowed the prisoners to escape. When the jailor saw this he intended to kill himself. Undoubtedly, this is because he knew he would have to take their place should they escape. But Paul called out and said in verse 28, "Do yourself no harm, for we are all here!" This shows that the guard did not want to take the place of the prisoners. In fact, in Acts 12:18-19 we read of how Herod ordered guards to be executed who had allowed Peter to escape.
Therefore, for the guards to risk their lives, a large bribe would be necessary. Who was more likely to have enough money to bribe the guards, the religious leaders or the disciples? Also, who had a greater motive, the disciples who wanted to have Jesus rise from the dead (risking further persecution), or the Jews who wanted to complete their attempt to be rid of Jesus?