Grammatically speaking, one must add the word "for" after the word "request". I'm requesting for a doctor. It is understood that his medical professional expertise is needed. It would be better to say "I'm requesting for the services of a doctor not a paramedic!"

" I request a doctor" I have to add that Michael is totally right,this sentence is not used in spoken English and would sound weird, but it is grammatically correct though,while the other one 'request to' is wrong. You could use 'request' in other forms: "I request something of a doctor" " I request something from a doctor" "I request that ,the doctor .........( continue according to context)."

'I request to the doctor' is completely wrong, however neither expression would be said in english. you would say - I need to see a doctor I need a doctor the word 'request' in this situation would sound very strange. a similar example - 'I request a policeman' we would never say that. Rather we would say 'I'm calling the police' - when you phone for the police or doctor you are in fact requesting they assist you, but you wouldn't actually use the word request.