I think you meant you updated the email address in your client's return but it's not showing up in Intuit Link. That's because Intuit Link pulls the data from the details saved under Clients instead of what's under Tax Returns and there's no synchronization between the two after the creation of the return.
To edit your client's particularly for Intuit Link, you will need to go to Clients, on the line for your client, click on the dropdown for View client & return, and select Edit client. Once you save the changes, the new email address will be reflected in Intuit Link and be used for future communication. When you send the request, Intuit Link may, however, prompt you whether this new email address instead of the existing one should be used. Just select the new email address and hit Continue.
If this is a client who already had responses and documents saved in Intuit Link previously, changing the email address will cause your client to lose access to those entries (although it won't affect yours). In case this applies to you, please come back as there is a different process to follow.
Still an AllStar