Learn Russian: Нам нужно срочно послать за врачом.

In Russian, when you move to fetch, retrieve, or pick up a person or object, you use за + the Instrumental case. If you used для врача, it would mean you are sending something for the doctor's personal benefit, rather than fetching the doctor to come help you.

We are saying 'to send for', but where is the word for the person we are actually sending to do the fetching?

You can leave it out entirely! Just like in English, you don't need to specify the messenger. The phrase послать за plus the Instrumental case is a standard set phrase when you need someone fetched, especially professionals like a doctor or the police.

Since нужно often triggers the imperfective aspect for general obligations, why do we use the perfective послать here?

While нужно can take an imperfective verb for general rules or repeated needs, we use the perfective послать here because it is a specific, one-time action that needs to be successfully completed right now. The word срочно (urgently) emphasizes this immediate, one-time necessity.

Why does врачом end in -ом instead of -ем? Doesn't ч usually take an е?