I have just done this for the privet plate that came on my car. The form is a V888 from the DVLA with a fee of between £2.50-£5.00. Depending on whether the car you are enquiring about is registered in your name or not.
The fact it is a “privet” plate makes no difference, it is still registered with the DVLA and they own it.
If the car/plate is in your name then they don’t seem to be that bothered about giving you the complete history of the car, all previous owners and addresses etc. only saying that the information provided is for your own personal records blah blah.
If the car is not registered in your name then as said above you need a
very valid reason for them to
consider giving it to you, this is usually in the case of an accident. I would defiantly give them a ring first and explain the situation and see what they say.
Also I believe they can also get in touch with the registered keeper to inform them that there has been an enquiry about the car/plate.
All the best in your endeavour.