Yes, it’s possible. The instructions are buried on one help page amongst dozens and you’re limited to downloading one at a time… but you can do it. Personally I’m just loathe to install a piece of software just to download one album every third blue moon, which is as often as I buy music from Amazon. One of the reasons for this is their insistence on making you install crapware just to get at the items you’ve legally purchased.
So, go to Amazon and purchase your album. Click the option to go to the Cloud Player not the one to download.
Locate and select one track to download by ticking its checkbox in the left hand column.
Click on the “Download (1)” button and you’re presented with the usual “Click this huge button to install the crappy download software” popup. Down at the bottom of this box is a little link to “Skip installation”. Click this instead.
You’ll now get your usual download dialogue allowing you to actually gain possession of the media you paid money for.
You can download more than one track at a time if you’re quick enough to uncheck one box, check the next one and click “Download (1)” again. This time you won’t get the annoying popup, instead just being presented with your web browser’s standard download dialogue.
Why Amazon have to make thing so bloody awkward, I don’t know. A simple “click to download ZIP” or similar would do the job. Regardless, I now have my free Nuclear Blast album downloaded so I can pop it on my phone and listen in the car. Happy now.