Martin,
Hope this isn't too late.
My husband and I have camped in Kenya on two occasions now. On both occasions we rented a Land Rover from Wildways. They are based in Karen and rent fully equipped Land Rovers (ie. all camping gear included).
Your booking includes a day's training etc.. to make sure that you are prepared for every eventuality.
We absolutely loved it and would recommend Wildways to any of our friends.
Their website is
http://www.wildways.biz
If you decide to contact them, say hi from Jess and Andy.
Jandica