I've simplified your letter a little. Also "Yours faithfully" is probably better when you are not addressing the person by name. I'm not sure that you need "an" before "early" but it's not a problem and OK if that's what you prefer.
Saturday, April 16, 2011
To: The General Manager
Sub: Request for an early annual leave.
I would like to submit a request for an early annual leave starting from 14th May 2011. The due date for my annual leave is 2nd June 2011.
Taking my leave from 14th May would enable me to spend time with my family during their annual vacation.
I hope this request will merit your approval.