I did some googling and came across an instruction page from the Vancouver library (not my library, just found this by chance) which indicates things may work slightly differently with different browsers.
http://guides.vpl.ca/sony-ereaders
Here's a quote:
Download eBooks to Your Computer and Transfer to your Sony eReader
•Select a format (EPUB format is recommended)
•Click the download button and open the downloaded file with the Sony for PC or Sony for Mac software
•The eBook will begin to download
•Depending on the Internet browser (Firefox, Internet Explorer, Safari or Chrome) that you use, the eBook may open automatically, or you may need to locate the .acsm file in the download folder on your computer,
◦Once you've located the .acsm file, right mouse click on it and select Open With from the menu
◦Choose Sony Reader for PC or Sony Reader for Mac as the program to open the file with
•Using a different browser can sometimes solve the issue if you have trouble opening the eBook or locating the .ascm file