I wouldn't solely trust each website's billing history. Some of them only keep their records available online for certain periods of time, some may be sold and bought by other companies and have data lost or unavailable in transition, and you will always need a net connection to access any of it.
You can download a free PDF printer such as Primo PDF, and go to each of your statements and convert them to .PDF files (if they aren't already available as such). You could go as far as paying for cloud storage and saving all of the PDFs both on your hard drive and on the cloud, or even just on a flash drive or external hard drive. The downside there is a similar risk as paper, if your residence were to be compromised, whether through fire, storm, theft, or computer failure, you may lose all your work.
The accountant in me would answer "absolutely not!" though, and point to PDF files if you're absolutely unwilling to keep paper around.