It's called SAYQUICK, you can use it to let your visitors to record a message.
Everything is done online. Nothing to download.
INBOX ... here you get the recorded message
and here you can hear it and download it
The problem is you still can't send a voice message. remember they are in beta. But I found the online solution really useful.
You can test it here:
Squadfree - Free bootstrap 3 one page template (just a dummy site)
SayQuick | Easy Voicemail for your Website
(I'm not affiliate to that site. Just thought it would be good to let you know about this.)