Call Handlers can never have their own voicemail box. If you set up a Call Handler to take a message, it must target some other object for the message itself.
If you set the Call Handler to "Take Message" and then set the target of the message to a distribution list with the folks in question, everyone will receive a copy and it should show up in each individual's Web Inbox.
If you check the box for "Dispatch Delivery" (under where you set the target of the Call Handler messages to a Distribution List) then everyone will get a copy of the message, but once one person has listened to it the message is withdrawn from the other voicemail boxes. I don't know if messages that are Dispatch Delivery will show up in Web Inbox, though.
Does this help?
Maren