It's not possible to move an MDF Request to a different Partner Organization, but you can manually recreate the Request (with the same information) under a new Organization using the duplicate functionality.
To duplicate an MDF Request, go to MDF > click on the Request to view more details > Actions > Duplicate.
MDF Request Details Page
The Duplicate MDF Request window will appear with the same information from the original Request. You'll want to update the MDF Request Owner field to a new Partner User from a different Partner Organization and click Save. 
Duplicate MDF Request Window
Once you've created the MDF Request under the new Partner Organization, you can delete the original Request under the other Partner Organization using the instructions in this article: How do I delete an MDF Request?