He can apply for DLA himself (or you can if you are still his representative until he turns 16). However, as DLA care component is not awarded because of having type 1 diabetes, but because of needing help from another person, the assumption is that a 16 year old can do everything himself, at least more than, say an 8 year old could. Very few adults with type 1 diabetes get DLA, either care or mobility components, unless they have complications eg their ability to see, need renal dialysis at home, have poor hypo awareness, problems getting around (eg periferal neuropathy). If you / he applies and are turned down, you can appeal and explain your case in person to a panel consisting of a legal chairman, medical doctor & person with experience of diabetes. A paper hearing before the same type of panel is far less likely to succeed, but in any case, the odds are against any adult with uncomplicated type 1 diabetes. Make sure you get a medical exemption card so he continues to get free prescriptions, due to having a qualifying condition after turning 16.
DLA forms are available online (use search engine - I haven't got time just now) or from Job Centres.