I was diagnosed at the age of 8, one of the reasons I was allowed out of hospital within 3 days is because I insisted on doing my own injections. The hospital and my parents had no issue with this, obviously there was always someone around when I took my injection. When I went onto 4 injections a day, my lunchtime one was always done in the school's medical room.
It's great that your son wants to start doing his own injections, it's a big responsibility and one he'll have for life, it's great that he's comfortable learning to do this on his own.
Having supervision is sensible, showing him what to do and check and seeing how much he can remember on his own is a great learning technique.