This applet will create the HTML code you need to add to your Tumblr page to enable a newsletter signup button.
Once you provide your MailChimp API key, you can select which list to use. The applet then will output the HTML code you need to paste into the "Info" section of your tumblr page (screenshots below)
You can then set up an RSS-to-Email MailChimp campaign to automatically send out your new Tumblr posts via email.
You can easily access your Tumblr RSS feed by appending RSS, e.g. http://testpages.tumblr.com/rss