Samson. I applied for a company that I was company pilot for a year ago. I believe it doesn't matter whether it is for a company or private, although you must apply using a company headed note paper and state your business. If you own an an aircraft, why not make that a business etc and apply. Or call yourself a company. I don't really know but one thing is for sure they didn't really care who I was provided I came up with a valid callsign request. I believe the service is still free. One of the last things in the aviation world that is. The callsign must be clear unambiguous and clean! Can't conflict with aviation phrases or be close to mistake. It makes finding a name a bitch. You will need at least three suggestions as I suspect the first two will be rejected. The company I flew for was only a shop!
Hope it helps.
Ps The guys name was Cyril Meadam - Callsigns AIS Heathrow