It might not work from Amazon's mobile site, but at the very bottom of the page is a link to "Amazon Full Site". Click it and you should be able to purchase without a problem. You may not be able to play the songs from your mobile device unless you have the Amazon Cloud Player app installed, but your purchase will be locked in and available for download at your convenience when you get back to your main computer.
You should not click each single - I believe that will charge you for each song individually, not getting you the deal. To the right of the photo of the album cover, click "Buy MP3 Album with 1-Click." You will then be walked through downloading the full album for the deal price.