Learn Russian: Надо вызвать врача.

The word надо means it is necessary or one must. It is an impersonal word, meaning it doesn't use a normal subject. Often, Russians leave out the pronoun if the person who needs to do the action is obvious from the situation. If you wanted to specifically say 'We need', you would use the dative case: нам надо. The English translation just adds 'We' to make it sound like a natural English sentence.

Doesn't the verb to call translate to позвонить? Why use вызвать here?

While позвонить means to call someone on the telephone for a conversation, вызвать means to summon someone or ask them to come to your location. When you need a doctor, an ambulance, the police, or a taxi to physically come to where you are, you must use вызвать.

Why does the word for doctor end in -а (врача) instead of being just врач?

The word врач is the direct object of the verb, so it must be in the accusative case. Because a doctor is a living being (this is called an animate noun), masculine nouns in the accusative case take the exact same ending as they do in the genitive case. For a masculine noun ending in a consonant like врач, you add -а to get врача.