As far as I know, MBNA require the 2nd user to be 18 but I don't have a vast array of credit cards so can't speak for all of them.
Have a look at somewhere like www.moneyfacts.co.uk which will list a variety of companies & call any numbers listed there to ask them.
Under different circumstances, Capital One seems to be in a hurry to throw money at people so I'd check them before some of the others.

Or, you might get lucky and have someone post an appropriate credit card company here.

I've had a second card on my parents Barclycard account since I was about 16, definitely under 18 as it was to be used for emegency transport home, or if we got into trouble when we first got our cars etc

Don't forget though, that credit cards (especially Barclays in my opinion!) WANT you to get into debt. The bigger your debt, the more interest they get from you. They are doing you no favours! They realise that 16 year olds are possibly more wayward than 18 year olds, and that will earn them more money. Go easy on it, especially when they offer to raise your credit level.