Records from Kenya after independence are one thing, but records kept while the British were in charge are quite another. Records kept by the British were moved to the U.K. and if I remember correctly they are stored at Oxford, and the British did keep meticulous records!
In Kenya, birth registration became compulsory for whites in 1904. For all birth, death and marriage certificates, you will need to contact the Department of Civil Registration in Nairobi.
If your ancestors were of British origin but were born in Kenya, the civil registration documents may possibly be available in the UK. The National Archives, Kew hold records relating to records kept by British consulates abroad and at sea. From the nineteenth century they were obliged to send the records that they recorded to the General Register Office of England and Wales. The army also kept its own registers for at home and abroad.
The National Archives also holds some records of birth, marriages and deaths in Kenya. Unfortunately, these are not complete, but what does exist is available in record class RG 36 and covers the years 1904 to 1924. The National Archives holds a microfilm index to these, which will make your search easier. The Kenyan index can be found in RG 43. You will need to visit the National Archives in person, as these records aren't available on the website. Source.
Historical records of the Kenyan Colonial Government may be stored at Oxford, of course.
Here's that article about research at the Kenya National Archives. Quite an experience.