Faire procéder à l’appel des témoins, c’est demander aux huissiers d’appeler et d’amener les témoins à la barre, devant la cour.

Ce n’est pas le président qui agit, mais il demande à un tiers d’appliquer la procédure de justice qui s’intitule l’appel des témoins (pour être placé à la barre des témoins).

faire + a verb + noun: in general. To have or make someone do something.

an exemple: Je vais faire nettoyer mon jardin. I am going to have my garden cleaned [by someone, passive] .

Mais le président a dit qu’il allait faire procéder à l’appel des témoins. But the judge said he was going to have the calling of witnesses started or begun [by someone].

in Larousse we see that procéder à/au quelque chose is: Faire quelque chose, l’exécuter selon un certain ordre, par une série d’opérations : Procéder au recensement des chômeurs. If you look in a good bilingual dictionary, it means: to start, initiate, or institute [as in institute proceedings]

Please note: I have laid out the sentences in English to mirror the French in order to demonstrate that this structure of faire + verb + à/au always suggests in French that someone else will be performing the action. It is not the judge him or herself who will call the witnesses. He is having them called, by a clerk of court or by having a summons issued. He is not directly calling them.

The Camus sentence can be translated several ways. That said, it should be passive:

The judge said he was proceeding with having the witnesses called. That is more idiomatic and keeps the passive structure intact.