Learn Portuguese (Portugal): Fizemos-lhe organizar o banquete.

This is the main rule for causative verbs like fazer (to make) and deixar (to let). Because the infinitive verb (organizar) already has a direct object (o banquete), the person being made to do the action (the causee) is pushed to the indirect object position. This prevents the sentence from having two direct objects at once.

What would the sentence look like if we didn't mention o banquete and just said "We made him organize"?

If the second verb drops its direct object, the person being forced to act shifts back to being the direct object. You would use the accusative pronoun o. Because Fizemos ends in an -s, that -s is dropped and the pronoun becomes -lo, resulting in Fizemo-lo organizar.

Could lhe also mean "her" or "you" in this sentence?

Yes. Because lhe is an indirect object pronoun, it can refer to him, her, or you (formal singular). Context usually makes it clear, but if it is ambiguous, a Portuguese speaker might opt for a different phrasing to explicitly state who is being forced to act.

Is there a way to phrase this without having to juggle indirect and direct objects?