In short, the answer to this question, if you are a member of the public, is no. Of course, there are several reasons why you might want to know who owns a number plate, from being involved in an accident to wanting to own a specific registration number but finding the registered keeper is not that simple.
What is in a Number?
The number plate belongs to the vehicle, so there is information available online to anyone that will tell you what car the plate belongs to, whether it is insured and taxed, etc. Now while this information might help if you have a prang, this is not a vigilante society so, if you are involved in an accident and have the registration number you need to seek help from the police or give the number to your insurance company, and they can take care of finding the person involved for you. If you are looking at purchasing a vehicle there are also paid checks you can run that will tell you if there is any outstanding finance on a car (HPI checks), this £25 check also shows you if the vehicle has been reported as lost or stolen and whether it has been involved in an accident and become any category of insurance write-off.
Sometimes people want to find out who owns a number plate because the random grouping of letters and numbers has a meaning for them and they would like to own the vehicle and or the number plate. It is possible to keep a number plate after scrapping the car it belonged to, but you will need to go through the correct DVLA channels. There are sites online that sell personalised plates which you can put on your current vehicle, but if you cannot find the one you really want, you have pretty much lucked out. If you happen to find the plate when the car is with the owner you could approach them about buying it, but personalised plates tend to be purchased for sentimental reasons, and people are not keen to sell them on.
There is a possibility of applying to DVLA for information about the owner of a registration plate, but you will have to have just cause. You will have to have exhausted all other channels (the police, leaving it to an insurance company etc.,) and you need to apply in writing. They may or may not accept your request but in cases where a vehicle has been abandoned for years on private land, or the car was used in insurance fraud etc. are potentially valid reasons but again if the police should be involved then you may not get far as this is for them to sort. Finally, if you are a member of either the International Parking Community or the British Parking Community because you issue parking tickets or notices of trespass, then you can apply for the register keeper details from DVLA. All of the application services have a charge which has to be paid.