For some reason, I always thought Doormaker and Clairvoyent remained together and were immobile stuck in a trance-like state doing their jobs... not sure why I thought that but I did.
Also, since Doormaker can remotely make a portal from one destination to another, he would not show up on Taylor's Snark-sense unless he personally was going in to pick it up. Those two combine made me believe Contessa asked for a portal to the box location, took the box then asked for a portal to the destination.
Even if Doormaker isn't immobile like I thought, allowing him to leave the Cauldron base to pick up / drop off packages is probably a risk. Especially since without him Cauldron gets very limited in their inter-dimensional travel. Not to mention Clairvoyant stays with him and never grew out of a child-mentality.
So ya, I can understand it not being Contessa, since she's got plenty of stuff to do, and picking up the boxes wouldn't need her Path telling her when it's ready since Clairvoyant can tell people when it's time, but I still think it'd be someone other than Doormaker.
Hmm... Another reason is Taylor's Snark sense would likely propagate on the other side of the portal and since Clairvoyant stays near Doormaker if it was him doing the pickup then she should be sensing two snarks not one.
Maybe it's the Custodian who does the pickup? Though she usually stays in the Cauldron base...