I actually just came across this add-on for doing this.
http://www.zen-cart.com/index.php?main_page=product_contrib_info&products_id=106
Download it and simply upload the three folders to your Zen-Cart root folder and replace all the files it shows already exist.
Then you log in to your Zen Cart admin area and under the "Customers" menu there will be a new item called "Subscription Manager". You can use this to add new customers to the newsletter email list.
You can also import a list of email address. I haven't played around with the different formats for importing but you ARE able to import from spreadsheets. I did a text from a TXT file with one email address on each line.
For address that are in comma delimited format enter a comma "," in the "Fields separated by" box. Do a small test file to make sure everything imports properly before importing your real email list. There is no way of undoing it without deleting each address one by one.
Instructions on how to import from different file formats are included in the "READ ME" file that comes with it.
Read the "READ ME" file before installing to make sure you understand how to import.
I hope this helps you because it helped me a lot! I was worried about installation but it installed very easily.