This is a rough idea.
Well: No permit needed, just dig a hole till you get water, its as wide as it gets. Depth about 20feet or thereabouts.
Borehole: Hole of about 250mm diameter drilled into soil formations untill an aquifer reached, depth can be upto 500m.
For a borehole, you will need a geologist who will do a test to determine soil formation and the expected depth of an aquifer. Based on the report, you will apply for a licence from WRMA(Water Resources Management Authority). Once you have the licence, you'll proceed to NEMA to get the ok.
Please note that most people prefer to have the Geologist handle all the above process of investigation and licencing.
Once the above documentation is complete, then and only then will you ask a Drilling Contractor to give you a quotation for the drilling works since you can approximate the aquifer depth.
The cost of drilling is about 5k~6.5k per metre depth.
Borehole drilling is a risky prospect, you might drill only to find water which is not of the right chemical composition, or simply find traces of mud.
If you are lucky to get good water(sample taken to Ministry of water/University/KEBS lab for analysis), you will need to do test pumping to determine the yield and quality of the borehole. It is the data obtained from the test pumping that will guide you in purchasing a submersible pump for the borehole. Dont know how much a pump costs but will ask hope to get back to you.
As for contractors who carry out drilling, i will try and get a list soon.
Best of luck.
Check on the thread below for more information.
http://m.wazua.co.ke/forum.aspx?g=posts&t=25