While requests cannot individually be moved to a different Organization, you can move a contact and with that contact, any requests attached to them will. This is commonly referred to as the boat method.
In this situation, we have Henry who is in organization A with his request, and we would like to move this request into organization B.
So we will move Henry over to Organization B, which will include all requests attached to his user. This is done from the Organization drop down on the edit user page.
Once Henry has moved over to Organization B, his request will also be in Organization B. At this point, you will want to move the Request to another user within Organization B, or you can make the request unassigned. This is done from the Reassign To drop down under Advanced Options on the Request Summary.
At this point, the request will no longer be connected to Henry. You can send Henry back to his original Organization A, and the request will remain in Organization B.