When I open the Reader app, there is a button at the bottom called "Store". When I click that it opens up a browser window with the address
http://ebookstore.sony.com. The webpage that it opens has a button called "My Account" where you can see your account information, wishlist, previous purchases, etc. When you find a book that you are interested in, the description page gives you the option to either "Add to Cart" or "Add to Wishlist".
I also have the Sony software installed on my computer since I have a Sony ereader. I wouldn't expect that to make a difference. Is your credit card up to date in your account info? Are you logged in to your account when you try to make a purchase? Maybe you have some setting in your browser preventing the transaction?
Sony at some point authorized my Android device to my Sony account because it shows up in my list of authorized devices. I don't recall doing anything special - maybe just entering in my account name and password when I initially installed the app?