Learn Italian: Chi vende i dolci? Li vendiamo noi. - Who is selling the desserts? We are selling them.

In Italian, when you want to emphasize WHO is doing the action, especially to answer a 'who' question, you put the subject pronoun after the verb. Saying Noi li vendiamo would sound flat, whereas Li vendiamo noi strongly emphasizes that WE are the ones selling them.

Could I keep the English word order and say 'Vendiamo noi li'?

No, unstressed object pronouns like li (them) must almost always go directly before the conjugated verb. Therefore, the structure has to be [object pronoun] + [verb] + + vendiamo + noi.

Why do we use li in the second sentence?

Li is the masculine plural direct object pronoun meaning 'them'. It replaces i dolci (the desserts) from the first question so you don't have to repeat the noun.

The English translation says 'We are selling them', but vendiamo is simple present. Could I use stiamo vendendo?

You could say Li stiamo vendendo noi if you want to stress that the action is happening right at this exact moment. However, the simple present (vendiamo) is frequently used in Italian for ongoing situations like running a stall or a business, whereas English naturally prefers the progressive form.