Audioboo allows visitors to record audio from their browser. The following line of code embeds an audioboo recorder button on your site [live example]:
Users would then have to choose to share their recording with you, perhaps by posting a link to the audioboo page in a form field or comment box.
Audioboo Pro accounts allow anonymous users to record audio directly to your own audioboo account, instead of to an anonymous audioboo user account, which may prove more useful to you because you'd have an automatic collection of all recordings. At present, you need to be a large organisation and email them to apply for a pro account, but they have plans for a 'plus' account for single users who want more powerful features.