I now travel without a laptop because my iPad does most things that I need. So when I wanted to download a book, I found that I ran into this problem as well and ended up finding an Amazon site that pointed me to the Kindle Store site:
https://www.amazon.com/gp/kindle/kcp/tos.html/ref=kin_ipad_glbl_surl_iksc#stos-b ookmark
I was able to then go to the site and buy the book I wanted which in turn downloaded immediately to my iPad. It does give you the option to "Read in Browser" but I clicked "Read Now" for the download.
The Amazon site I got the Kindle link from was found in the third (#3) FAQ titled, "How do I buy books on Kindle for iPad or Kindle for iPhone?" and the web address is:
http://www.amazon.com/gp/help/customer/display.html?nodeId=200298460
I hope this helps.