At the moment, Sony does not support the Mobipocket/AZW format used by the Kindle store, and you cannot buy Kindle e-books from Amazon without owning a Kindle reader (or an iPhone with a Kindle app). So I'm afraid you (and a lot of other people in similar situations) are out of luck getting Kindle books into your Sony.
Your best bet is to go to other e-book sites and try to find the e-books in venues other than Amazon. Many publishers will offer their books on their own websites or other e-book sites, if they've made them available to Amazon. Even Mobipocket formatted e-books can often be converted to the Sony format, or to ePub, with third-party software (like Calibre, mentioned in the forums area in this site).