Learn Italian: Facciamo costruire il tavolo al falegname. - We're having the carpenter build the table.

This is the core rule of the causative construction! When the infinitive verb (costruire) has a direct object (il tavolo), the person actually doing the action is pushed into being an indirect object. That is why we use a + il = al. It is almost like saying: we cause the building of the table to the carpenter.

What would happen if we didn't mention the table at all?

If there is no direct object in the sentence, the person performing the action remains the direct object. You would drop the preposition a and simply say Facciamo costruire il falegname (We make the carpenter build). The a is only added to prevent the sentence from having two direct objects at the same time!

How would I replace the carpenter with a pronoun to say we are having HIM build the table?

Because the carpenter is acting as an indirect object here, you must use the indirect object pronoun gli (to him). Object pronouns go right before the conjugated verb, so the sentence becomes: Gli facciamo costruire il tavolo.

Can this construction also mean we are letting him build it?

Yes! Using fare + infinitive generally means to have or make someone do an action, but it can also mean to let or allow them to do it. Depending on the situation, this exact sentence can also translate as we are letting the carpenter build the table.