Login to Action Network and go to the group you'd like the sign up link for.
Under Actions, click Forms
The screen below will come up, and you'll fill in the highlighted sections.
Click Save And Go To Next Step.
The screen below will come up next, and you'll edit the "Thanks for your support" and "Help us meet our goal by spreading the word about this action using the tools on this page" sections to what you'd like them to be.
Click Save & Publish
The highlighted link on the right is the link you can share for people to join your group's Action Network email list.
If you'd like to tag the people who specifically sign up through this link, click Edit Tags, highlighted above. Then, click the plus sign and make a new tag called "Email Sign Up Link" or something similar that works for your group.