Visual Voicemail does not have this feature. I know that Google Voice (and possibly Youmail?) will send an e-mail with the transcript and recording of the voicemail, as well as make them accessible online. They are also free to use, which is a plus over Verizon's product.
Just to add to spottedcatfish's response - YouMail DOES send an email with a recording, and you can log in and manage your voicemail messages online WITHOUT having the phone with you.
And to Mrs. Jake - you don't have to listen to each message entirely to delete it when calling in .... just press "7" (or whatever the delete button is for VVM) as soon as you are sure you don't need the message, and move on to the next.