Question Restatement, part 1: Were the Guards that guarded Jesus' tomb Pilate's Guards, or the Jews? (Conclusion at Bottom.)
Question Restatement, part 2: Is there an indication, in the Greek Syntax, that Pilate may have said, "Have a guard", (Imperative Mood), rather than "You already have a guard", (Indicative Mood)? To show that Pilate sent his own Guards?
Why Did the Jews Ask Pilate for a Guard?
Although the Jews certainly had a guard of their own, (Herod's, (Luke 23:11), and the Temple Guards, (Acts 4:1, Acts 5:23)), they probably would have asked Pilate for the help anyway because it was a High Holy Day, a Sabbath--Passover.
NOTE: The day that the Passover Lamb is Sacrificed, and then eaten later that night--is always a Sabbath, (Ex. 12:6), no matter what day of the week it falls on. That is why it reasonable to conclude that Jesus was Crucified during the day on Thursday--because any day would be a Sabbath, a high day, (John 19:31), if it was when the Passover was to be Sacrificed and eaten.
"Have" is Certainly Indicative, and not in the Imperative Mood
It has already been noted in another proposed answer that the word "Have" in Greek, "Ἔχετε", can be either Second Person Indicative, (a statement of fact), or Imperative, (a command).
So then, "How do you tell the difference in the text?" By the Context of course ...
Rather than ride that "Bandwagon" of relying on traditional translations, consider tackling the question from scratch, and doing the research ourselves:
Of the ~51 occurrences of "Have", (A.) in the New Testament, (B.) in the second person, Present, and Plural; (C.) and at first glance could either be Indicative or Imperative; --Only just FOUR passages express "Have" in the Imperative, and we KNOW these cases are in the Imperative mood, rather than Indicative, because the Context CLEARLY demands and requires it. There is no such requirement in Matthew 27:65, and to force an Imperative sense in this passage is not well-ground upon any linguistic basis :
Clear Illustrations of "Have" in the "Imperative" case.
Mark 9:50 V-PMA-2P
GRK: αὐτὸ ἀρτύσετε ἔχετε ἐν ἑαυτοῖς
NAS: will you make it salty [again]? Have salt
Mark 11:22 V-PMA-2P
GRK: λέγει αὐτοῖς Ἔχετε πίστιν θεοῦ
NAS: saying to them, Have faith
Philippians 2:29 V-PMA-2P
GRK: τοιούτους ἐντίμους ἔχετε
NAS: joy, and hold men like
James 2:1 V-PMA-2P
GRK: ἐν προσωπολημψίαις ἔχετε τὴν πίστιν
NAS: My brethren, do not hold your faith
ἔχετε: "You all - have"
51 Occurrences in the NT
8 Occurrences in Matthew
ἔχει: "He - have"
104 Occurrences in the NT
13 Occurrences in Matthew
ἔχεις: "You - Have"
28 Occurrences in the NT
1 Occurrence in Matthew
So, Whose Guard were They?
Conclusion: In the end, the Guard clearly belonged to the Jews, either because Pilate assigned the Guard to them, or because the Jews already had them.
The origin of where the Guard came from cannot be argued conclusively, as the text does not clarify the origin of the Guard, just the end result.
Personally, from the Narrative, I am persuaded that it was probably a Temple Guard, or Herod's own.
Authority Over the Guards:
Regardless of where the guards came from, the Jews were ordered to secure the Tomb, and exercised authority over the guards:
Matt. 27:65, NASB - Pilate said to them, "You have a guard; go, make it as secure as you know how."
The Guard Remained During the Sabbath
As Passover is always a Sabbath, no matter the day of the week it falls on, it is likely the guards remained Thursday afternoon/evening until Sunday morning.
It could be argued that Jewish Guards would not have done this, and would have asked Pilate to use his guards -- but even this isn't "conclusive", because there were obvious exceptions to Sabbath "work" in view of military service, etc, (i.e., 2 Kings 11:5, etc.).
**Matt. 28:4 - ** The guards shook for fear of him and became like dead men.
The Guard Reported to the Chief Priests--Not Pilate
The idea that Roman Guards would have reported to the Jews first, rather than Pilate, and only "some" of them, is compelling. Whether they were from Pilate, or the Jews, they risked dying either way, (Jewish authority killing incompetent guards: Acts 12:19).
Matt. 28:11 - Now while they were on their way, some of the guard came into the city and reported to the chief priests all that had happened.
The Guard was Bribed, affirming they were not Roman Guards
Speculating that the guards, if Roman, feared Pilate finding out about a "stolen body/resurrection", more than they feared Pilate finding out about being bribed -- makes no sense, at all. However, Jewish guards being paid to give false testimony about Jesus' disciples, makes a lot more sense. Though, still inconclusive.
No matter whose guards they were, Pilate probably would have been upset because his orders were disobeyed. But, because Pilate was reluctant to kill Jesus anyway, Pilate would not have likely cared if his body was stolen.
Matt. 28:12 - And when they had assembled with the elders and consulted together, they gave a large sum of money to the soldiers, 13 and 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.”