Learn Russian: Как зовут вашего сына?

In Russian, asking someone's name literally translates to "How do they call..." rather than "What is...". Because of this, you must use the question word как (how) when asking about names with the verb зовут.

Why is the verb зовут in the "they" form when we are asking about just one son?

This is an impersonal sentence. Instead of saying "What is his name?", Russian effectively says "How do they call him?". The "they" is a general, unnamed group, so the verb takes the third-person plural form зовут (they call), even though the topic of the question is just one person.

Why is the phrase вашего сына in this specific case?

Because the literal translation is "How do they call your son?", the phrase "your son" is the direct object receiving the action of the verb зовут. Therefore, it must be in the accusative case. Since a son is an animate masculine noun, its accusative form looks exactly like the genitive case, adding -а to make сына and changing ваш to вашего.

How would I use this sentence in indirect speech, like "She asked what your son's name is"?

To turn this into an indirect question, you simply keep the question word как and use the exact same sentence structure after the main clause. For example: Она спросила, как зовут вашего сына. Notice that unlike English, the word order inside the reported Russian question does not change at all.