If I am the registered keeper of my car and I am a named driver on the insurance certificate, can I still tax it?

Course you can. As long as there's a valid MOT certificate for the vehicle and the insurance is up-to-date, you shouldn't have any problems getting your road tax sorted. However it's worth pointing out that putting someone else as a main driver to bring insurance premiums down is referred to as fronting in the insurance field and this usually invalidates your policy if your insurer finds out. I'll be more worried about this than the road tax to be honest. Are you the principal user of the car?