You'll be provided with a webpage version of all of your sent emails. To find that link, go to the email page and click on share URL (next to the email's open rate).
I can completely understand where you're coming from, having an archived list tool is a great feature to have for your customers- and luckily we offer the ability to have that feature here at Constant Contact! We offer an Embeddable Newsletter Archive Widget lets you host up to 100 of your previously sent emails on your website. Visitors to your website can catch up on any emails they may have missed, and if you put the widget close to the sign-up form on your website, you may even get a few new subscribers to your email list. Here is a great resource where you can read more about this feature, Add the embeddable Newsletter Archive widget to the source code of a website and choose the maximum ...
Along with our Newsletter Archive widget, every single one of your sent emails has it's own permanent URL for you to share. Both these features will allow you to give your contacts the chance to see your past newsletters. Please reach out if you have any additional questions!