Currently there's no way to access the Chat Channels panel to configure voice chat in ElvUI. Well, I created a WeakAura that remedies this.
Right-clicking the little "page" icon in any of your chat tabs (excluding the "Combat Log" tab) will show a chat menu. The WeakAura I made adds a "Chat Channels" option to show/hide the Chat Channels UI panel.
IMPORTANT: You need to reload your UI after importing the aura for the first time.