Red Lodge karting over near Newmarket have this that might be of interest?
http://www.redlodgekarting.com/topkatz.html
They also run Non-MSA series and you would be surprised how cheap you could pick up a Honda Cadet or such for that. Give them a shout.
There are club only championships for younger kids - I know Coast2Coast karting down in Hayle where Im from run a championship so I expect there will be one somewhere near you.