Learn Icelandic: Þarft þú stóla?

Because it is the direct object of the verb þurfa (to need), which forces it into the accusative case. For a masculine noun like stóll (chair), the nominative plural is stólar (chairs), but the accusative plural drops the -r to become stóla.

How would this change if I was asking "Do you need the chairs"?

You would attach the accusative plural definite ending to the noun. For masculine nouns, you add -na to the bare accusative plural form, making it stólana. The full sentence would become Þarft þú stólana?

Why does the verb þurfa (to need) look so different as Þarft?

Þurfa is an irregular verb. When paired with þú (you, singular), it takes the -t ending and also changes its root vowel from u to a, becoming þarft.

Why doesn't Icelandic use a word like "Do" to ask this question?

Icelandic doesn't use helping verbs like "do" to form questions. Instead, it simply swaps the order of the subject and the verb. While a statement is Þú þarft (You need), a question is just Þarft þú? (Need you?).