I use the stock email app for my work, hotmail, and a home email account. For all of those I have the "Postman" chime set for notification, and vibrate turned off. When I use the main sound setting on the phone and switch to either mute or vibrate mode, I do not get any sound or vibration when emails come in to those email accounts.
I also use the Gmail app for a Gmail account. Although I have similar settings ("Postman" sound notification but the "vibrate" box in the gmail notifications settings is NOT checked), I still get a vibrate notification (but no sound) when the phone's main sound setting is set to "vibrate." When the phone is set to "mute," I do not get either sound or vibration when a gmail arrives.
You might want to try using the stock email app for your yahoo and gmail accounts and see what happens.