A Yahoo! Group has two primary features: an email list and a collection of web resources. To subscribe to the mailing all you need is an email address. To access the web resources, including the forms to change the email address used for your subscriptions, you'll need an account at Yahoo so that you can sign in.
If you don't have, and don't wish to obtain, a Yahoo Account your only option is to unsubscribe the old address and subscribe the new address. That has the virtue of simplicity, but doing so abandons any content (messages) you've contributed to those groups, and any elevated status (such as moderator) you may have been given. Yahoo policy is that any content you contribute to Yahoo Groups remains your property, and that you may remove such content at any time[1]. To unsubscribe and re-subscribe by email, use the Email Commands. You will find them listed on the Info/About page of the group, see the "Visit Your Group" link at the bottom of any Individual or Digest message from the group.
NOTE: The Yahoo help page Activities available via email in Yahoo Groups fails to mention that Yahoo will send an email in response, and you must receive and respond to that email before the command takes effect. Archived page.
If you don't yet have a Yahoo account, but wish to preserve your ownership of content, or your status in groups, then it is important to complete the change of address Via Web (below) before you lose access to the old email address.
Note: over time Yahoo! has made changes to the procedures to change email addresses. So understand that any instructions you find, even Yahoo's own help pages, may contain obsolete information or may be missing needed information. This page was created during an instance where the existing Yahoo help pages contained obsolete information[3]. If you find any error or inaccuracy in this information you can help by writing or correcting items, or just by commenting. Please read About YahooGroupedia.
After that is done messages from this group should come to your newly selected email address, and you will be able to post messages to the group from the new address.
1. Yahoo Terms of Service, paragraph 9: "Yahoo does not claim ownership of Content you submit..." and 9a: "This license exists only for as long as you elect to continue to include such Content on the Yahoo Services and will terminate at the time you remove or Yahoo removes such Content from the Yahoo Services."
2. As of early November, 2015, the Yahoo help page Change your personal email address in a group, step 3, refers to an "Add New Email" link which no longer exists. Archived page.