Are the two organizations sharing the same itemset? If they are not sharing an ITEMSET it is not possible. If they are using the same item set, there are some MAXVAR settings that dictate whether a PO is required for internal transfers across organizations (or sites). Go to the Organization application and open each Organization. Under the Inventory Options choose Transfer Options.
A shipment is always required across organizations. A PO is not required by default but can be made required for a transfer across Organizations.
Assuming a PO is not required, you should be able to create an inventory usage record with the from site & storeroom matching the originating storeroom. On the line set the desired target site and storeroom. You will need security access to both the originating and destination site and storeroom to do this. Change the inventory usage to SHIPPED which will enable you to create a shipment. Then in the Shipment Receiving go through the process of receiving the shipment.