I don't know if there's any easy way to export the list of contact email addresses, but there's a way you can get them. If you find the Contacts link and go to that, you then need to scroll past all of the groups you might have and you'll come to the list of your frequent contacts. There is a link to view all contacts, and that is the link you want.
That link takes you to a page with all of the addresses you've ever sent an email to, and again you'll have to navigate past all of your groups ... the fastest way to do that is to press t to jump to the first table, which contains all of those contacts. The table contains two columns, the first being the contact name, and the second being the contact's email address. I don't know if there is a way to select just the email address column, but that would be the fastest way if anybody knows how. What I'd do is use NVDA's speech viewer (in NVDA's Tools menu) and then go to your contact table and press CTRL+ALT+Down Arrow, and NVDA will speak the email address as you press the down arrow. The Speech Viewer window will record what has been spoken, so just keep pressing CTRL+ALT+down arrow until you get to the end of the list, and then go to the Speech Viewer window and select the first to the last address, and copy and paste it wherever you want it.
You might want to jump to the bottom of the table first to see how many contacts you have. I have over 1800, so that is 1800 CTRL+ALT+Down Arrow presses in order to record all of them in the Speech Viewer window. If I had no other way of doing it, and if I needed it, that's the way I'd do it :)
Hope this helps, and hopefully somebody else might have a speedier method.