Learn Russian: Они хотят успеть заполнить эти документы в среду. - They want to manage to fill out these documents on Wednesday.

This lesson is about надо and нужно. How would I change this sentence to say "They need to fill out these documents" instead of "They want to..."?

You would use the Dative case for "they" (им) and the impersonal word надо or нужно. The sentence would become: Им надо успеть заполнить эти документы в среду.

Why are there two infinitive verbs in a row (успеть заполнить)?

In Russian, just like in English, you can chain verbs together. The first verb, хотят (want), is conjugated to match the subject они (they). Any following verbs attached to that same thought stay in their basic infinitive form: успеть (to manage or have time) and заполнить (to fill out).

Does успеть just mean "to manage"?

Успеть specifically means "to have enough time to do something" or "to make it in time". It is slightly different from just succeeding at a difficult task. Here, it means they want to make sure they have the time to get the documents filled out on Wednesday.

What case is эти документы in?

It is in the Accusative case because it is the direct object of the verb заполнить (the thing being filled out). For inanimate plural nouns like this one, the Accusative form looks exactly the same as the Nominative (dictionary) form.