After Basic Training and going directly to WOCS (Ft. Rucker), do the candidates get to have their POV immediately?

Last student packet from WOCS -
https://usacac.army.mil/sites/default/files/documents/cace/wocc/WOCS_Orientation_Packet_20200901.pdf

9. PRIVATELY OWNED VEHICLES. During in-processing you will park your POV in a
designated class area, where it will remain for the duration of your time in WOCS. Store
valuable personal items in the security room, not in your POV. Every Sunday, you will be
allowed to start and idle your vehicle to prevent battery and engine problems. Unless
approved by the HHC XO or Primary TAC Officer, you are not authorized to move your
vehicle once training begins. Ensure you have a current driver’s license, vehicle
registration, and proof of insurance. Those documents cannot expire during or immediately
after the course. Keep the documents inside your POV.

Yes, but not accessible during the 5 week duration other than to start them weekly

Not while a candidate in the academy. They will park their POV's and have access to them once per week only to start them, so the battery doesn't die. Once graduated, obviously, they will get them back.